जेल प्रहरी 2020 के लिए आपको बिल्कुल मुफ्त टेस्ट सीरीज उपलब्ध कराई जा रही है नीचे लिंक पर क्लिक करें और मध्यप्रदेश सामान्य प्रश्नोत्तरी में स्वयं का आकलन करें
Friday, 18 September 2020
Tuesday, 11 September 2018
Monday, 6 August 2018
तर्क का बीजगणित
तर्क का बीजगणित
तर्कशास्त्र ,तर्क प्रक्रिया के नियमों मे अंतर्गत िदिए गए कथनों से मान्य निष्कर्ष पर पहुॅचने का एक िविशेष तकनीकी रूप है।
तर्क वह माध्यम और रूप प्रदान करता है जिससे मनुष्य द्वारर सोचने की प्रकिया व्यवस्थित क्रम से हाेे सके अर्थात समस्या के हल का माध्यम प्राप्त हो सकेा वैसे तर्क की प्रकिया सभी िविषयों पर लागू होती है,परन्तु गणित में हम िनिरंतर अपने बनाए हुए कथनों की उपपत्त्यिॉ देते रहते है।उपपत्त्यिॉ देने की प्रकिया पूर्णत: तार्किक होती है। इन प्रकियओं के अध्ययन का विषय तर्कशास्त्र है।
प्रस्तुत अध्याय में हम प्रतीकात्मक तर्कशास्त्र का प्रारंभिक अध्ययन करेंगे िजिसके अंतर्गत प्रतीकाेेंेंऔर गणिता की संक्रियाओं का अध्ययन तथा िविशलेषण िकिया जाता है। इस अध्ययन का मूल उद्देश्य गणितीय उपपत्त्िा की व्याख्या करना है।जो वास्तव में गणितीय कथनों कस सउद्देश्य अनुक्रम है।इसके िलिए तार्किक वाक्यों एवं तार्किक संयोजकों की सहायता ली जाती है। िजिसे हम गणित की भाषा कहते है।
1840 में िब्रिटिश गणितज्ञ डीीमार्गन ने तर्क के िविकास पर कार्य िकिया।डीीमार्गन भारत में पैदा हुए थे तथा इ्रग्लैण्ड में उनकी िशिक्षा हुई।1847 में डी मागन के कार्य के बाद में जार्ज बूल ने एक पुस्तक प्रकाशित की िजिसका शीर्षक था the methemarical analysis of logic!
यह ग्रंथ तर्कशास्त्र में प्रयुक्त गणितीय िविश्लेषण था। लगभग एक शताब्दी तक यह अन्वेषण गुमनामी केे अंधेरे में रहा।सन 1938 में क्लाडे ई.शेनन ने यह बताया िकि बूलीय बीजगणित के अनेक िनियमों का उपयोग टेलीफोन जाल में सरलीकरण के अत्यंत्ा महत्वपूर्ण है।सरलीकरण की इस प्रकिया का उपयोग अब कम्प्यूटर के परिपथों के सरलीकरण में भी होने लगा है ।इस प्रकार यह बीजगणित अत्यंत महत्वपूर्ण हो गया ।
बूलीय बीजगणित व्यापक रूप से (1,0) फलन के रूप में जानी जाती है।पहले इनका उपयोग उन कथनों में िकिया जाता था जो सत्य या असत्य होते थे परंतु वर्तमान में इसका उपयोग िस्विचन परिपथ ,जो या तो खुले हों या बंद हों में िकिया जाता है।
बूलीय बीजगणित में मुख्यत: तीन संक्रियाऍ होती है- (1) 'तथा' (2) 'अथवा') (3) 'नही' िजिन्हें प्रतीक रूप में क्रमश: v एवं ^ से िलिखेंगे।इन प्रतीकों को क्रमश: हम '+' , '.' या ' ' ' भी िलखेंगे।
तर्कशास्त्र ,तर्क प्रक्रिया के नियमों मे अंतर्गत िदिए गए कथनों से मान्य निष्कर्ष पर पहुॅचने का एक िविशेष तकनीकी रूप है।
तर्क वह माध्यम और रूप प्रदान करता है जिससे मनुष्य द्वारर सोचने की प्रकिया व्यवस्थित क्रम से हाेे सके अर्थात समस्या के हल का माध्यम प्राप्त हो सकेा वैसे तर्क की प्रकिया सभी िविषयों पर लागू होती है,परन्तु गणित में हम िनिरंतर अपने बनाए हुए कथनों की उपपत्त्यिॉ देते रहते है।उपपत्त्यिॉ देने की प्रकिया पूर्णत: तार्किक होती है। इन प्रकियओं के अध्ययन का विषय तर्कशास्त्र है।
प्रस्तुत अध्याय में हम प्रतीकात्मक तर्कशास्त्र का प्रारंभिक अध्ययन करेंगे िजिसके अंतर्गत प्रतीकाेेंेंऔर गणिता की संक्रियाओं का अध्ययन तथा िविशलेषण िकिया जाता है। इस अध्ययन का मूल उद्देश्य गणितीय उपपत्त्िा की व्याख्या करना है।जो वास्तव में गणितीय कथनों कस सउद्देश्य अनुक्रम है।इसके िलिए तार्किक वाक्यों एवं तार्किक संयोजकों की सहायता ली जाती है। िजिसे हम गणित की भाषा कहते है।
1840 में िब्रिटिश गणितज्ञ डीीमार्गन ने तर्क के िविकास पर कार्य िकिया।डीीमार्गन भारत में पैदा हुए थे तथा इ्रग्लैण्ड में उनकी िशिक्षा हुई।1847 में डी मागन के कार्य के बाद में जार्ज बूल ने एक पुस्तक प्रकाशित की िजिसका शीर्षक था the methemarical analysis of logic!
यह ग्रंथ तर्कशास्त्र में प्रयुक्त गणितीय िविश्लेषण था। लगभग एक शताब्दी तक यह अन्वेषण गुमनामी केे अंधेरे में रहा।सन 1938 में क्लाडे ई.शेनन ने यह बताया िकि बूलीय बीजगणित के अनेक िनियमों का उपयोग टेलीफोन जाल में सरलीकरण के अत्यंत्ा महत्वपूर्ण है।सरलीकरण की इस प्रकिया का उपयोग अब कम्प्यूटर के परिपथों के सरलीकरण में भी होने लगा है ।इस प्रकार यह बीजगणित अत्यंत महत्वपूर्ण हो गया ।
बूलीय बीजगणित व्यापक रूप से (1,0) फलन के रूप में जानी जाती है।पहले इनका उपयोग उन कथनों में िकिया जाता था जो सत्य या असत्य होते थे परंतु वर्तमान में इसका उपयोग िस्विचन परिपथ ,जो या तो खुले हों या बंद हों में िकिया जाता है।
बूलीय बीजगणित में मुख्यत: तीन संक्रियाऍ होती है- (1) 'तथा' (2) 'अथवा') (3) 'नही' िजिन्हें प्रतीक रूप में क्रमश: v एवं ^ से िलिखेंगे।इन प्रतीकों को क्रमश: हम '+' , '.' या ' ' ' भी िलखेंगे।
बलीय बीजगणित का प्रत्येेक फलन िजिसमें अचर न हो को ,एक िवियोजनीय प्रसामान्य रूप में व्यक्त कर सकते है।
बलीय बीजगणित का प्रत्येेक फलन िजिसमें अचर न हो को ,एक िवियोजनीय प्रसामान्य रूप में व्यक्त कर सकते है।
उप्पत्ति-
परिभाषा से हम जानते है कि फलन का िवियोजनीय प्रसामान्य रूप चरों के गुणनफलों के योग के रूप में प्रदर्शित होता है।
माना f(x1,x2.....xn) बूलीय बीजगणित B(+,.,') मेंं n चरों x1,x2.... xn का एक अचर रहित फलन है ।प्रथम चरण मेें व्यंजक या फलन से ( यदि हो तो) डी-मार्गन िनियम का प्रयोग कर पूरक (') को खाेलते है। िद्वतीय चरण में ,यदि आवश्यक हो ,तो वितरण नियम का प्रयोग कर (.) का (+) पर िवितरण करते है।
अब तृतीय चरण में एक चर माना xi के साथ अन्य चर लाने के िलिए xi को xi1लिखते है। तत्प श्चात् xi1 के स्थान पर xi(xj+xj') लिखते हैा जिससे फलन में xixj+xixj' गुणन के योग में आते है।
इसी प्रक्रिया को दोहराकर सभी चर x या x'के रूप में उपस्थित रहते है!
अंतिम रूप से अब पुनरावृत्त्ि वाले पदों काेे केवल एक बार (वर्गसम िनियम से)लिखा जाता है। इस प्रकार से िदिया गया फलन िवियोजनीय प्रसामान्य रूप मेें परिवर्तित होता है।
उप्पत्ति-
परिभाषा से हम जानते है कि फलन का िवियोजनीय प्रसामान्य रूप चरों के गुणनफलों के योग के रूप में प्रदर्शित होता है।
माना f(x1,x2.....xn) बूलीय बीजगणित B(+,.,') मेंं n चरों x1,x2.... xn का एक अचर रहित फलन है ।प्रथम चरण मेें व्यंजक या फलन से ( यदि हो तो) डी-मार्गन िनियम का प्रयोग कर पूरक (') को खाेलते है। िद्वतीय चरण में ,यदि आवश्यक हो ,तो वितरण नियम का प्रयोग कर (.) का (+) पर िवितरण करते है।
अब तृतीय चरण में एक चर माना xi के साथ अन्य चर लाने के िलिए xi को xi1लिखते है। तत्प श्चात् xi1 के स्थान पर xi(xj+xj') लिखते हैा जिससे फलन में xixj+xixj' गुणन के योग में आते है।
इसी प्रक्रिया को दोहराकर सभी चर x या x'के रूप में उपस्थित रहते है!
अंतिम रूप से अब पुनरावृत्त्ि वाले पदों काेे केवल एक बार (वर्गसम िनियम से)लिखा जाता है। इस प्रकार से िदिया गया फलन िवियोजनीय प्रसामान्य रूप मेें परिवर्तित होता है।
Tuesday, 24 July 2018
percentage
percentage:-
100% = 1
90% =9/10
80% = 80/100 =4/5
70% = 70/100 =7/10
60% = 60/100 =3/5
50% = 50/100 =1/2
40% = 40/100 = 2/5
30% = 30/100 =3/10
25% = 25/100 =1/4
20% = 20/100 =1/5
10% =10/100 =1/10
other form
1/2 =50%
1/3 =33.33%
1/4 =25%
1/5 =20%
1/6 =16.66%
1/7 =14.28%
1/8 =12.5
1/9 =11.11%
1/10 =10%
1/11 =9.99%
If the price of a commodity increases by a% then the reduction in sonsumption so as not to increase the expenditure is:
( p/p+100)*100
question
1. A batsman scored 110 runs which included 3 boundaries and 8 sixes what percent of his total score did he make by running between the wickets?
A. 45% b 45 5/11 % c. 54 6/11% D 55%
adding soon...
100% = 1
90% =9/10
80% = 80/100 =4/5
70% = 70/100 =7/10
60% = 60/100 =3/5
50% = 50/100 =1/2
40% = 40/100 = 2/5
30% = 30/100 =3/10
25% = 25/100 =1/4
20% = 20/100 =1/5
10% =10/100 =1/10
other form
1/2 =50%
1/3 =33.33%
1/4 =25%
1/5 =20%
1/6 =16.66%
1/7 =14.28%
1/8 =12.5
1/9 =11.11%
1/10 =10%
1/11 =9.99%
If the price of a commodity increases by a% then the reduction in sonsumption so as not to increase the expenditure is:
( p/p+100)*100
question
1. A batsman scored 110 runs which included 3 boundaries and 8 sixes what percent of his total score did he make by running between the wickets?
A. 45% b 45 5/11 % c. 54 6/11% D 55%
adding soon...
Sunday, 22 July 2018
Multification 11 with any number of three digits
1.
213*11 = take 2 then take 2 and 1 and add them then take 1 and 3 and add them and then take 3
= so 2 (2+1) (1+3) 3
= so ans is 2343
another example is
352*11 = 3 (3+5) (5+2) 2
= so ans is 3872
give your feed here
213*11 = take 2 then take 2 and 1 and add them then take 1 and 3 and add them and then take 3
= so 2 (2+1) (1+3) 3
= so ans is 2343
another example is
352*11 = 3 (3+5) (5+2) 2
= so ans is 3872
give your feed here
Wednesday, 6 June 2018
square of numbers
NUMBER SQUARE
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400
21 441
22 484
23 529
24 576
25 625
26 676
27 729
28 784
29 841
30 900
42 1764
43 1849
44 1936
45 2025
46 2116
47 2209
48 2304
49 2401
50 2500
1 1
2 4
3 9
4 16
5 25
6 36
7 49
8 64
9 81
10 100
11 121
12 144
13 169
14 196
15 225
16 256
17 289
18 324
19 361
20 400
21 441
22 484
23 529
24 576
25 625
26 676
27 729
28 784
29 841
30 900
31 961
41 1681
32 1024
33 1089
34 1156
35 1225
36 1296
37 1369
38 1444
39 1521
40 1600
42 1764
43 1849
44 1936
45 2025
46 2116
47 2209
48 2304
49 2401
50 2500
DOWNLOAD it
Tuesday, 22 May 2018
fast calculation trick
multiple 99*56
now we can multiple 99 with any number an easy way like this
first we take 56 .
now we write 55 (one digit less than every number)
then we substract it also with 99 ie 99-55=44
so ans is 5544
multiple 999*545
similarly
we first we take 545
and we do one digit less from 545 ,so it is 543
and also substract 543 from 999 ,so it is 999-543=654
so ans is 543654
for video click here
now we can multiple 99 with any number an easy way like this
first we take 56 .
now we write 55 (one digit less than every number)
then we substract it also with 99 ie 99-55=44
so ans is 5544
multiple 999*545
similarly
we first we take 545
and we do one digit less from 545 ,so it is 543
and also substract 543 from 999 ,so it is 999-543=654
so ans is 543654
for video click here
Sunday, 20 May 2018
FAST CALCULATION TRICK
FAST CALCULATION TRICK
Q- square of 43
We take two base 25 and 50
1. difference from 25 and 43 is 18
2 difference from 50 and 43 is 7 ,so square of 7 is 49
and the ans is 1849
For video CLICK HERE
Q- square of 43
We take two base 25 and 50
1. difference from 25 and 43 is 18
2 difference from 50 and 43 is 7 ,so square of 7 is 49
and the ans is 1849
For video CLICK HERE
Thursday, 17 May 2018
FAST CALCULATION OF SQUARE TRICK
welcome to SMOKEMATHS ,
now you can watch the trick of fast calculation of fast trick on youtube . its a easy way to find the square of any number.so you can just click here and watch on youtube the fast calculation of square trick
FOR VIDEO CLICK HERE
now you can watch the trick of fast calculation of fast trick on youtube . its a easy way to find the square of any number.so you can just click here and watch on youtube the fast calculation of square trick
FOR VIDEO CLICK HERE
Wednesday, 16 May 2018
FIND THE SQUARE OF ANY NUMBER EASY WAY:-
Now we take a number ,square of 52
we take a method like this
Now we take a number ,square of 52
we take a method like this
first we take it like
we multiple first two number 2 and 2 =4 like
now we do cross multiple of two digit and then add like 2*5+5*2=20
and take it 0 and 2 carry
we take 0 and carry 2
now we mumtiple last two digit and add carry like 5*5+2=27
so ans is 2704
ANOTHER METHOD IS
WE CHOSE A BASE OF 25
52 IS MORE THEN 27 FROM 25 SO WE TAKE =27
AND THE SQUARE OF 2 IS 04, SO WE TAKE 04
SO ANS IS 2704
Sunday, 13 May 2018
Wednesday, 20 September 2017
PROBABILITY ( प्रायिकता )
PROBABILITY- in 1993 A.N. Kolmogrov, a Russian mathematician tried succesfully to relate the theory of probability eith the set theory by axiomatic aproch.
Random variables - A real valued function defined on a sample -space is called a randon - variable .
SAMPLE SPACE- A sample space of a random experiment is the set of all possible outcomes of that experiment and is denoted by S.
for example- if a coin is tossed then there are two possibilities either we shall get a head or tail . we denoted here head (H) and tail(T).
EXHAUSTIVE EVENTS- All possible outcomes in a trial are called exhaustive events.
SAMPLE POINT:- Every element of the sample space is called a sample point. and it it is contains finite number of point then its is called finite sample point.for example in a coin H and T.
EVENTS:- Of all the possible outcomes in the sample space of an experiment some outcomes satisfy a specified description , it is called an event and its denoted by E.
CERTAINS AND IMPOSSIBLE EVENTS:- if S is a sample space then S and Φ are both subset of S and both are events.so S is called certains events and Φ is called impossible events.
EQUALLY LIKELY EVENTS:- Two wvents are considered equally likely if one of them cannot ve expected in preference to the other.
EXHAUSTIVE EVENT:- All possible outcomes in a trial are called exhaustive events.
for example :- if we trail a coin then the exhaustive events are H and T.
MUTUALLY EVENT OR INCOMPITIBLE EVENT:- Two or more than two evetns are called Mutually exclusive events if there are no element common to these events .
if E1 and E2 are two mutually exclusive events then E1 =E2 =Φ
EXAMPLES :- An experiment in which two coins are tossed together fine the sample space.
solve- S={(H,H) (H,T) (T,T) (T,H)}
IF coins tossed tree times then S ={ (HHH, TTT, THH, HHT, HTT, THT,}
SAMPLE OF COMPOUND EVENTS:- If E contains only one elements of the sample space S then E is called simple events i.e E= {ei}
compound events :- If E contains more than one elements of the sample space S then S is called compound event.
E={ei} where i = 1,2,3....n
FORMULLE FOR PROBABILITY OF AN EVENT:- let E be an event of S containing m element of S ie n(E) = m if P(E) is the probability of the event E happening then P(E) = n(E) /n(S)
COMPLEMENTARY EVENT:- if E be an event then not happining of the event E is called the complementary event of E and is denoted by E'.
and p(E) + p(E') = 1
Q:- find the probability of throwing on even number with a die.
solve :- let S be a sample space and the event of getting an event number be E then
S ={ 1,2,3,4,5,6} and E={2,4,6}
so n(S) = 6 and n(E) = 3 [ for even number]
therefore , P(E) = n(E) /n(S)
= 3/6
= 1/2
Q:- if two coin tossed , find the chance that there should be heads on both.
solve:- if two coins tossed then sample space is
S={(H,H) (H,T) (T,T) (T,H)}
both head have only one condition that is ( H,H)
therefore P(E) = n(E) /n(S)
= 1/4
Q:- Find the probability of throwing on even number with die.
solve:-let S be the sample space and the event of getting an even number be E then
S = {1,2,3,4,5,6} and E={2,4,6}
n(S) = 6 and n(E) = 3
therefore the probability of event of happening p(E)=n(E)/n(S)
= 3/6
=1/2 =ans
COMPOSITION OF EVENTS:-
THE EVENT REPRESENTED BY AUB OR A+B:-- IF the event E happens when A happens or B happens then E is denoted by and E is represented by AUB i.e. E=AUB
THE EVENT REPRESENTED BY A ∩ B OR AB:--If The event E happens when the events A and B both happens then the events E is represented by A ∩ B OR AB i.e. E=A ∩ B
COMPLEMENT OF EVENT A OR THE EVENT A':-- If the event E happpens when the event A does not happen then E is denoted by A'
Theoram :- If E1 and E2 are any two events then P(E1 U E2) = P(E1)+P(E2)-P(E1∩ E2).
PROOF:-
lets S be the sample space and n be the number of elements in the events in S.
let l be the number of elements in E1 and m the number of elements in the events E2
i.e. n(S)= n n(E2)= l n(E2)=m
if the events E1 and E2 are not mutually exclusive then the E1∩ E2 is not equal to phy.
let n(E1∩ E2) = r
clearly, n(E1U E2) = l+m-r
now the probability of E1 and E2 happening denoted by P(E1∩ E2) is given by
p(E1U E2) =
Random variables - A real valued function defined on a sample -space is called a randon - variable .
SAMPLE SPACE- A sample space of a random experiment is the set of all possible outcomes of that experiment and is denoted by S.
for example- if a coin is tossed then there are two possibilities either we shall get a head or tail . we denoted here head (H) and tail(T).
EXHAUSTIVE EVENTS- All possible outcomes in a trial are called exhaustive events.
SAMPLE POINT:- Every element of the sample space is called a sample point. and it it is contains finite number of point then its is called finite sample point.for example in a coin H and T.
EVENTS:- Of all the possible outcomes in the sample space of an experiment some outcomes satisfy a specified description , it is called an event and its denoted by E.
CERTAINS AND IMPOSSIBLE EVENTS:- if S is a sample space then S and Φ are both subset of S and both are events.so S is called certains events and Φ is called impossible events.
EQUALLY LIKELY EVENTS:- Two wvents are considered equally likely if one of them cannot ve expected in preference to the other.
EXHAUSTIVE EVENT:- All possible outcomes in a trial are called exhaustive events.
for example :- if we trail a coin then the exhaustive events are H and T.
MUTUALLY EVENT OR INCOMPITIBLE EVENT:- Two or more than two evetns are called Mutually exclusive events if there are no element common to these events .
if E1 and E2 are two mutually exclusive events then E1 =E2 =Φ
EXAMPLES :- An experiment in which two coins are tossed together fine the sample space.
solve- S={(H,H) (H,T) (T,T) (T,H)}
IF coins tossed tree times then S ={ (HHH, TTT, THH, HHT, HTT, THT,}
SAMPLE OF COMPOUND EVENTS:- If E contains only one elements of the sample space S then E is called simple events i.e E= {ei}
compound events :- If E contains more than one elements of the sample space S then S is called compound event.
E={ei} where i = 1,2,3....n
FORMULLE FOR PROBABILITY OF AN EVENT:- let E be an event of S containing m element of S ie n(E) = m if P(E) is the probability of the event E happening then P(E) = n(E) /n(S)
COMPLEMENTARY EVENT:- if E be an event then not happining of the event E is called the complementary event of E and is denoted by E'.
and p(E) + p(E') = 1
Q:- find the probability of throwing on even number with a die.
solve :- let S be a sample space and the event of getting an event number be E then
S ={ 1,2,3,4,5,6} and E={2,4,6}
so n(S) = 6 and n(E) = 3 [ for even number]
therefore , P(E) = n(E) /n(S)
= 3/6
= 1/2
Q:- if two coin tossed , find the chance that there should be heads on both.
solve:- if two coins tossed then sample space is
S={(H,H) (H,T) (T,T) (T,H)}
both head have only one condition that is ( H,H)
therefore P(E) = n(E) /n(S)
= 1/4
Q:- Find the probability of throwing on even number with die.
solve:-let S be the sample space and the event of getting an even number be E then
S = {1,2,3,4,5,6} and E={2,4,6}
n(S) = 6 and n(E) = 3
therefore the probability of event of happening p(E)=n(E)/n(S)
= 3/6
=1/2 =ans
COMPOSITION OF EVENTS:-
THE EVENT REPRESENTED BY AUB OR A+B:-- IF the event E happens when A happens or B happens then E is denoted by and E is represented by AUB i.e. E=AUB
PROOF:-
lets S be the sample space and n be the number of elements in the events in S.
let l be the number of elements in E1 and m the number of elements in the events E2
i.e. n(S)= n n(E2)= l n(E2)=m
if the events E1 and E2 are not mutually exclusive then the E1∩ E2 is not equal to phy.
let n(E1∩ E2) = r
clearly, n(E1U E2) = l+m-r
now the probability of E1 and E2 happening denoted by P(E1∩ E2) is given by
p(E1U E2) =
hence proved
Tuesday, 22 August 2017
GRAPH
We study here some type of graph like multigraph ,multigraph,path,circuit,etc...
now we define here a graph. start as...
DIRECTED GRAPH:- A directed graph is a graph defined abstractly as an ordred pair (V,E) where V={v1,v2,.....} vertex and E={e1,e2,e3.........} i.e. egdes is a binary relation on V. The pair (vi,vj) is said to be incedent vertices.where vi is a initial vertax and vj is a terminal vertix.
SELF LOOP:- The defination of a graph an edge to be of the form (vi,vj) suvh an edge having the same vertex as both its end vertices is called a self loop
where K is a self loop
PARALLEL EDGES IN A GRAPH:- Let G=(V,E) is a graph then all edges having the same pair of end vertices are called parallel edges ..
where e1 and e2 are parallel edges.
UNDIRECTED GRAPH:- An undirected graph is defined abstractly as an ordered pair (V,E) where V is a non-empty set and E is a multisets of two elements from V.for example
this is an undirected graph because it have not any direction.
SIMPLE GRAPH:- A graph G=(V,E) that has neither self -loop nor parallel edges is called a simple graph. for example
it have no self loop and no parallel edges.
FINITE AND INFINITE GRAPH:- A graph with a finite number of vertices as well as a finite number of edges is called a finite graph otherwise it is infinite graph.
this is a finite graph because it have finite number of vertices and edges.
this is an infinite graph.
ORDER OF A GRAPH:- If G=(V,E) is a fiinite group then the number of vertices is is called the order of the graph G and its denoted by IVI (mod V).
INCEDENCE :- Let E be and edge joining two vertices vi to vj of a graph G=(V,E) then the edge e is said to be incident on each of its end vertices vi to vj.
ADJACENCY:- Two vertices in a graph are said to ve adjacent if there are exist an edge joining the vertices.
DEGREE OF A VERTEX:- The degree of a vertex v in a graph written as d(v) is equal to the number of edges which are incident on v with self -loop counted twice.
In fig A d(a)=2 , d(b)=2 , d(C)= 2
ISOLATED VERTEX:- a vertex in a graph G having no edge incedent on it is called an isolated vertex.
in this graph f is a isolated vertex.
PENDANT VERTEX:- A vertex v in a graph G is said to ve pendent vertex if its degree is one. i.e. d(v) = 1 . in fig B degree of e is 1.
NULL GRAPH:- A graph is said to be null if its each vertex is zero degree or other word its vertices are non empty but its edes are empty.
EVEN OR ODD VERTICES:- A vertex is said to be even or odd if its degree is an even of odd number.
THEORAM:- The sum of the degrees of all vertices in a graph G is equal to twice the number of edges in G.
solve:- Let G=(V,E) be a graph ,then the number of edges in Gis IEI,
since the edge in G is incident on two vertices so it contributes 2 to the sum of the degrees of the graph ,
so sum of degrees of all the vertices in G = ∑d(v)=2IEI
this theoram is called the HANDSHAKING LEMMA.
FOR EXAMPLE:-
now we define here a graph. start as...
DIRECTED GRAPH:- A directed graph is a graph defined abstractly as an ordred pair (V,E) where V={v1,v2,.....} vertex and E={e1,e2,e3.........} i.e. egdes is a binary relation on V. The pair (vi,vj) is said to be incedent vertices.where vi is a initial vertax and vj is a terminal vertix.
SELF LOOP:- The defination of a graph an edge to be of the form (vi,vj) suvh an edge having the same vertex as both its end vertices is called a self loop
where K is a self loop
PARALLEL EDGES IN A GRAPH:- Let G=(V,E) is a graph then all edges having the same pair of end vertices are called parallel edges ..
UNDIRECTED GRAPH:- An undirected graph is defined abstractly as an ordered pair (V,E) where V is a non-empty set and E is a multisets of two elements from V.for example
SIMPLE GRAPH:- A graph G=(V,E) that has neither self -loop nor parallel edges is called a simple graph. for example
 |
| FIG A |
FINITE AND INFINITE GRAPH:- A graph with a finite number of vertices as well as a finite number of edges is called a finite graph otherwise it is infinite graph.
this is a finite graph because it have finite number of vertices and edges.
ORDER OF A GRAPH:- If G=(V,E) is a fiinite group then the number of vertices is is called the order of the graph G and its denoted by IVI (mod V).
INCEDENCE :- Let E be and edge joining two vertices vi to vj of a graph G=(V,E) then the edge e is said to be incident on each of its end vertices vi to vj.
ADJACENCY:- Two vertices in a graph are said to ve adjacent if there are exist an edge joining the vertices.
DEGREE OF A VERTEX:- The degree of a vertex v in a graph written as d(v) is equal to the number of edges which are incident on v with self -loop counted twice.
In fig A d(a)=2 , d(b)=2 , d(C)= 2
ISOLATED VERTEX:- a vertex in a graph G having no edge incedent on it is called an isolated vertex.
 |
| fig B |
PENDANT VERTEX:- A vertex v in a graph G is said to ve pendent vertex if its degree is one. i.e. d(v) = 1 . in fig B degree of e is 1.
NULL GRAPH:- A graph is said to be null if its each vertex is zero degree or other word its vertices are non empty but its edes are empty.
THEORAM:- The sum of the degrees of all vertices in a graph G is equal to twice the number of edges in G.
solve:- Let G=(V,E) be a graph ,then the number of edges in Gis IEI,
since the edge in G is incident on two vertices so it contributes 2 to the sum of the degrees of the graph ,
so sum of degrees of all the vertices in G = ∑d(v)=2IEI
this theoram is called the HANDSHAKING LEMMA.
FOR EXAMPLE:-
Friday, 18 August 2017
RELATION : BINARY RELATION AND POSET
RELATION:- In order to express a relation from set A to Set B , we always need a statement which connects the elements of A with the elements of B.
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc
BINARY RELATION -
let A and B be two non-empty sets . A relation from A to B is a subset of A-B and is denoted by R .R is a relation from A to B such that R ⊆ A-B
R = {(a,b) : a∈A , b∈B and aRb}
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc that is R ⊆ A-B we can write this like R ⊆ AxB
memory :- if Set A and B has m and m elements that we write it total number of elements is m.m or squre of m.
if Set A has m and set B has n elements than we write is m.n elements
INVERSE RELATION:-if R ⊆ AxB is a relation from A to B then inverse relation define as R-1= { (b,a):(a,b)∈A a∈A , b∈B and aRb}
COMPOSITE RELATION:- Let A , B,C be three non-empty sets and R be a relation from A to B and S be a relation from B to C st R ⊆ AxB and S ⊆ BxC then the composite relation of two relations is a relation from A to C define as SoR ={(a,b) : an element c∈B st (a,c)∈R , (c,b)∈S}
i.e. (a,c)∈R , (c,b)∈S=> (a,b)∈ SoR .
Rusult:- if R-1 and S-1 are inverse of the relations R and S respectively then
(RoS)-1 = S-1 o R-1
RELATION ON A SET :- Let R be a relation from P to Q . if P =Q then we say that R is a relation on a Set P. i.e R ⊆ PxQ.
TYPES OF BINARY RELATIONS:-
1. Reflexive:- Let A be a non- empty set and R be a binary relation in A i.e. R ⊆ AxA then the relation R is called relexive relation if every element of A is R related to itself.
i.e. for all (a,a)∈R
2. Symmetric :-
If R is a relation in the set A then R is called symmetric relation if a is related to a then a is related to b.
i.e. (a,b)∈R => (b,a)∈R.
3. Anty-symmetric Relation :-
If R is a relation in the set A , then B is called anty-symmetric if (a,b)∈R (b,a)∈R. => a=b
4. Transitive Relation :-
If R is a relation in the set A then R is called transitive if a is related to b and b related to c then a is related to c i.e.
(a,b)∈R (b,c)∈R =>(a,c)∈R.
Partial Order set (poset):- A relation R on a set A is said to be Partial order set if it is Reflexice , Antisymmtric and Transitive. And if it satisfies another axiom trichnotomy low then it is called total ordering relation. trichnotomy law is any two element of set is comparble.
compleltely defination of Partial order set is :- A set A with a partial ordering relation≤on A is called partial order set and its denoted by( A,≤)
for Example:- If N be the set of partial integers then prove that relation ≤ is a partial order set on N.
solve:-
1. Reflexivity:- for each a∈N . since a is less or equal to itself then we can say a≤a. i.e. aRa.
2. Anti-symmtry:- let a,b∈N such that a≤b , b≤a so no element cannot be less or equal each other. therefore a=b.
3.Transitivity:- let a,b,c∈N st a≤b , b≤c then a≤c.
and hence it is a partial order set. hence proved.
Maximal element:- let ( A,≤) be a partially order set .An element a in A is called maximal element if
Minimal element:- let ( A,≤) be a partially order set .An element p in A is called maximal element if x≤ p => m=x.
Least Upper Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) x≤a for all x∈S.
2) there are exist a Upper Bound t such that a≤t.
I.e. lub is less or equal all other upper bounds.
Greatest lower Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) a≤x for all x∈S.
2) there are exist a lower Bound w such that w≤a.
I.e. glb is greater or equal all other lower bounds.
LATTICE:- A partially ordered set ( L,≤) is said to be lattice if evety two elements in the set L has a unique least upper bound and a unique greatest lower bound or its inf or sup exist.
DUAL LATTICE:- let ( L,≤) be a partial order set for any two element a,b∈L the converse of the relation "≤ " denoted by "≥" defined as b≥a <=>a≤b . in other words ( L,≤) is a lattice then ( L,≥) also a lattice.
continue..
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc
BINARY RELATION -
let A and B be two non-empty sets . A relation from A to B is a subset of A-B and is denoted by R .R is a relation from A to B such that R ⊆ A-B
R = {(a,b) : a∈A , b∈B and aRb}
example ;-A ={1,2,3,4} B={2,4,6} and relation is less than than we express it like (1,2) (1,4) (1,6) (2,4) (2,6) (3,4) (4,6). that is 1 is lass than 2 , 1 is less than 4 ...etc that is R ⊆ A-B we can write this like R ⊆ AxB
memory :- if Set A and B has m and m elements that we write it total number of elements is m.m or squre of m.
if Set A has m and set B has n elements than we write is m.n elements
INVERSE RELATION:-if R ⊆ AxB is a relation from A to B then inverse relation define as R-1= { (b,a):(a,b)∈A a∈A , b∈B and aRb}
COMPOSITE RELATION:- Let A , B,C be three non-empty sets and R be a relation from A to B and S be a relation from B to C st R ⊆ AxB and S ⊆ BxC then the composite relation of two relations is a relation from A to C define as SoR ={(a,b) : an element c∈B st (a,c)∈R , (c,b)∈S}
an element c∈B st (a,c)∈R , (c,b)∈S}i.e. (a,c)∈R , (c,b)∈S=> (a,b)∈ SoR .
Rusult:- if R-1 and S-1 are inverse of the relations R and S respectively then
(RoS)-1 = S-1 o R-1
RELATION ON A SET :- Let R be a relation from P to Q . if P =Q then we say that R is a relation on a Set P. i.e R ⊆ PxQ.
TYPES OF BINARY RELATIONS:-
1. Reflexive:- Let A be a non- empty set and R be a binary relation in A i.e. R ⊆ AxA then the relation R is called relexive relation if every element of A is R related to itself.
i.e. for all (a,a)∈R
2. Symmetric :-
If R is a relation in the set A then R is called symmetric relation if a is related to a then a is related to b.
i.e. (a,b)∈R => (b,a)∈R.
3. Anty-symmetric Relation :-
If R is a relation in the set A , then B is called anty-symmetric if (a,b)∈R (b,a)∈R. => a=b
4. Transitive Relation :-
If R is a relation in the set A then R is called transitive if a is related to b and b related to c then a is related to c i.e.
(a,b)∈R (b,c)∈R =>(a,c)∈R.
Partial Order set (poset):- A relation R on a set A is said to be Partial order set if it is Reflexice , Antisymmtric and Transitive. And if it satisfies another axiom trichnotomy low then it is called total ordering relation. trichnotomy law is any two element of set is comparble.
compleltely defination of Partial order set is :- A set A with a partial ordering relation≤on A is called partial order set and its denoted by( A,≤)
for Example:- If N be the set of partial integers then prove that relation ≤ is a partial order set on N.
solve:-
1. Reflexivity:- for each a∈N . since a is less or equal to itself then we can say a≤a. i.e. aRa.
2. Anti-symmtry:- let a,b∈N such that a≤b , b≤a so no element cannot be less or equal each other. therefore a=b.
3.Transitivity:- let a,b,c∈N st a≤b , b≤c then a≤c.
and hence it is a partial order set. hence proved.
Maximal element:- let ( A,≤) be a partially order set .An element a in A is called maximal element if
Minimal element:- let ( A,≤) be a partially order set .An element p in A is called maximal element if x≤ p => m=x.
Least Upper Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) x≤a for all x∈S.
2) there are exist a Upper Bound t such that a≤t.
I.e. lub is less or equal all other upper bounds.
Greatest lower Bound:-
Let S is a subset of R. And a∈R. Then a is called least upper bound of S if it is satisfied two axioms.
1) a≤x for all x∈S.
2) there are exist a lower Bound w such that w≤a.
I.e. glb is greater or equal all other lower bounds.
LATTICE:- A partially ordered set ( L,≤) is said to be lattice if evety two elements in the set L has a unique least upper bound and a unique greatest lower bound or its inf or sup exist.
DUAL LATTICE:- let ( L,≤) be a partial order set for any two element a,b∈L the converse of the relation "≤ " denoted by "≥" defined as b≥a <=>a≤b . in other words ( L,≤) is a lattice then ( L,≥) also a lattice.
continue..
Monday, 14 August 2017
Vector space
Vector space:-
Let V be a set with operation + and let F be a field with the operations + and . (dot) . an algebric expression ((V,+),(F,+.), .) with the internal and external operations is called vector space if it is satisfies following axioms.
1. (V,+) be an abelian group.
2. (F,+,.) Be closer with respect to dot.
3. a(æ+ß) = aæ+aß for all a,b is in F and æ,ß in V.
In other words we can express it as below..
we can describe this
1. (V,+) be an abelian group i,e, it satisfies 5 charactiristic closure associate identity inverse and commutative /
1 α,β in V then α+β in V.
2 α,β,γ in V then (α+β)+γ= α+(β+γ)
3 α in V , then there are exist 0 st α+0 = α
4 if α in V then -α in V st α+(α) in V.
5 α,β in V then α+β =β+α
2. (F,+,.) Be closer with respect to dot. i,e,
a in F and α in V st aα in V.
* set of real no is a vector space .
Let V be a set with operation + and let F be a field with the operations + and . (dot) . an algebric expression ((V,+),(F,+.), .) with the internal and external operations is called vector space if it is satisfies following axioms.
1. (V,+) be an abelian group.
2. (F,+,.) Be closer with respect to dot.
3. a(æ+ß) = aæ+aß for all a,b is in F and æ,ß in V.
In other words we can express it as below..
we can describe this
1. (V,+) be an abelian group i,e, it satisfies 5 charactiristic closure associate identity inverse and commutative /
1 α,β in V then α+β in V.
2 α,β,γ in V then (α+β)+γ= α+(β+γ)
3 α in V , then there are exist 0 st α+0 = α
4 if α in V then -α in V st α+(α) in V.
5 α,β in V then α+β =β+α
2. (F,+,.) Be closer with respect to dot. i,e,
a in F and α in V st aα in V.
* set of real no is a vector space .
Thursday, 3 August 2017
Useful formula Geometry formulla (उपयोगी सूत्र )
Geometry formula
(α в ¢)²= α² в² ¢² 2(αв в¢ ¢α)
1. (α в)²= α² 2αв в²
2. (α в)²= (α-в)² 4αв b
3. (α-в)²= α²-2αв в²
4. (α-в)²= f(α в)²-4αв
5. α² в²= (α в)² - 2αв.
6. α² в²= (α-в)² 2αв.
7. α²-в² =(α в)(α - в)
8. 2(α² в²) = (α в)² (α - в)²
9. 4αв = (α в)² -(α-в)²
10. αв ={(α в)/2}²-{(α-в)/2}²
11. (α в ¢)² = α² в² ¢² 2(αв в¢ ¢α)
12. (α в)³ = α³ 3α²в 3αв² в³
13. (α в)³ = α³ в³ 3αв(α в)
14. (α-в)³=α³-3α²в 3αв²-в³
15. α³ в³ = (α в) (α² -αв в²)
16. α³ в³ = (α в)³ -3αв(α в)
17. α³ -в³ = (α -в) (α² αв в²)
18. α³ -в³ = (α-в)³ 3αв(α-в)
ѕιη0° =0
ѕιη30° = 1/2
ѕιη45° = 1/√2
ѕιη60° = √3/2
ѕιη90° = 1
¢σѕ ιѕ σρρσѕιтє σƒ ѕιη
тαη0° = 0
тαη30° = 1/√3
тαη45° = 1
тαη60° = √3
тαη90° = ∞
¢σт ιѕ σρρσѕιтє σƒ тαη
ѕє¢0° = 1
ѕє¢30° = 2/√3
ѕє¢45° = √2
ѕє¢60° = 2
ѕє¢90° = ∞
¢σѕє¢ ιѕ σρρσѕιтє σƒ ѕє¢
2ѕιηα¢σѕв=ѕιη(α в) ѕιη(α-в)
2¢σѕαѕιηв=ѕιη(α в)-ѕιη(α-в)
2¢σѕα¢σѕв=¢σѕ(α в) ¢σѕ(α-в)
2ѕιηαѕιηв=¢σѕ(α-в)-¢σѕ(α в)
ѕιη(α в)=ѕιηα ¢σѕв ¢σѕα ѕιηв.
» ¢σѕ(α в)=¢σѕα ¢σѕв - ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв ѕιηαѕιηв.
» тαη(α в)= (тαηα тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1 тαηαтαηв)
» ¢σт(α в)= (¢σтα¢σтв −1) / (¢σтα ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв 1) / (¢σтв− ¢σтα)
» ѕιη(α в)=ѕιηα ¢σѕв ¢σѕα ѕιηв.
» ¢σѕ(α в)=¢σѕα ¢σѕв ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв ѕιηαѕιηв.
» тαη(α в)= (тαηα тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1 тαηαтαηв)
» ¢σт(α в)= (¢σтα¢σтв −1) / (¢σтα ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв 1) / (¢σтв− ¢σтα)
α/ѕιηα = в/ѕιηв = ¢/ѕιη¢
» α = в ¢σѕ¢ ¢ ¢σѕв
» в = α ¢σѕ¢ ¢ ¢σѕα
» ¢ = α ¢σѕв в ¢σѕα
» ¢σѕα = (в² ¢²− α²) / 2в¢
» ¢σѕв = (¢² α²− в²) / 2¢α
» ¢σѕ¢ = (α² в²− ¢²) / 2¢α
» Δ = αв¢/4я
» ѕιηΘ = 0 тнєη,Θ = ηΠ
» ѕιηΘ = 1 тнєη,Θ = (4η 1)Π/2
» ѕιηΘ =−1 тнєη,Θ = (4η− 1)Π/2
» ѕιηΘ = ѕιηα тнєη,Θ = ηΠ (−1)^ηα
1. ѕιη2α = 2ѕιηα¢σѕα
2. ¢σѕ2α = ¢σѕ²α − ѕιη²α
3. ¢σѕ2α = 2¢σѕ²α − 1
4. ¢σѕ2α = 1 − ѕιη²α
5. 2ѕιη²α = 1 − ¢σѕ2α
6. 1 ѕιη2α = (ѕιηα ¢σѕα)²
7. 1 − ѕιη2α = (ѕιηα − ¢σѕα)²
8. тαη2α = 2тαηα / (1 − тαη²α)
9. ѕιη2α = 2тαηα / (1 тαη²α)
10. ¢σѕ2α = (1 − тαη²α) / (1 тαη²α)
11. 4ѕιη³α = 3ѕιηα − ѕιη3α
12. 4¢σѕ³α = 3¢σѕα ¢σѕ3α
» ѕιη²Θ ¢σѕ²Θ=1
» ѕє¢²Θ-тαη²Θ=1
» ¢σѕє¢²Θ-¢σт²Θ=1
» ѕιηΘ=1/¢σѕє¢Θ
» ¢σѕє¢Θ=1/ѕιηΘ
» ¢σѕΘ=1/ѕє¢Θ
» ѕє¢Θ=1/¢σѕΘ
» тαηΘ=1/¢σтΘ
» ¢σтΘ=1/тαηΘ
» тαηΘ=ѕιηΘ/¢σѕΘ continue...
(α в ¢)²= α² в² ¢² 2(αв в¢ ¢α)
1. (α в)²= α² 2αв в²
2. (α в)²= (α-в)² 4αв b
3. (α-в)²= α²-2αв в²
4. (α-в)²= f(α в)²-4αв
5. α² в²= (α в)² - 2αв.
6. α² в²= (α-в)² 2αв.
7. α²-в² =(α в)(α - в)
8. 2(α² в²) = (α в)² (α - в)²
9. 4αв = (α в)² -(α-в)²
10. αв ={(α в)/2}²-{(α-в)/2}²
11. (α в ¢)² = α² в² ¢² 2(αв в¢ ¢α)
12. (α в)³ = α³ 3α²в 3αв² в³
13. (α в)³ = α³ в³ 3αв(α в)
14. (α-в)³=α³-3α²в 3αв²-в³
15. α³ в³ = (α в) (α² -αв в²)
16. α³ в³ = (α в)³ -3αв(α в)
17. α³ -в³ = (α -в) (α² αв в²)
18. α³ -в³ = (α-в)³ 3αв(α-в)
ѕιη0° =0
ѕιη30° = 1/2
ѕιη45° = 1/√2
ѕιη60° = √3/2
ѕιη90° = 1
¢σѕ ιѕ σρρσѕιтє σƒ ѕιη
тαη0° = 0
тαη30° = 1/√3
тαη45° = 1
тαη60° = √3
тαη90° = ∞
¢σт ιѕ σρρσѕιтє σƒ тαη
ѕє¢0° = 1
ѕє¢30° = 2/√3
ѕє¢45° = √2
ѕє¢60° = 2
ѕє¢90° = ∞
¢σѕє¢ ιѕ σρρσѕιтє σƒ ѕє¢
2ѕιηα¢σѕв=ѕιη(α в) ѕιη(α-в)
2¢σѕαѕιηв=ѕιη(α в)-ѕιη(α-в)
2¢σѕα¢σѕв=¢σѕ(α в) ¢σѕ(α-в)
2ѕιηαѕιηв=¢σѕ(α-в)-¢σѕ(α в)
ѕιη(α в)=ѕιηα ¢σѕв ¢σѕα ѕιηв.
» ¢σѕ(α в)=¢σѕα ¢σѕв - ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв ѕιηαѕιηв.
» тαη(α в)= (тαηα тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1 тαηαтαηв)
» ¢σт(α в)= (¢σтα¢σтв −1) / (¢σтα ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв 1) / (¢σтв− ¢σтα)
» ѕιη(α в)=ѕιηα ¢σѕв ¢σѕα ѕιηв.
» ¢σѕ(α в)=¢σѕα ¢σѕв ѕιηα ѕιηв.
» ѕιη(α-в)=ѕιηα¢σѕв-¢σѕαѕιηв.
» ¢σѕ(α-в)=¢σѕα¢σѕв ѕιηαѕιηв.
» тαη(α в)= (тαηα тαηв)/ (1−тαηαтαηв)
» тαη(α−в)= (тαηα − тαηв) / (1 тαηαтαηв)
» ¢σт(α в)= (¢σтα¢σтв −1) / (¢σтα ¢σтв)
» ¢σт(α−в)= (¢σтα¢σтв 1) / (¢σтв− ¢σтα)
α/ѕιηα = в/ѕιηв = ¢/ѕιη¢
» α = в ¢σѕ¢ ¢ ¢σѕв
» в = α ¢σѕ¢ ¢ ¢σѕα
» ¢ = α ¢σѕв в ¢σѕα
» ¢σѕα = (в² ¢²− α²) / 2в¢
» ¢σѕв = (¢² α²− в²) / 2¢α
» ¢σѕ¢ = (α² в²− ¢²) / 2¢α
» Δ = αв¢/4я
» ѕιηΘ = 0 тнєη,Θ = ηΠ
» ѕιηΘ = 1 тнєη,Θ = (4η 1)Π/2
» ѕιηΘ =−1 тнєη,Θ = (4η− 1)Π/2
» ѕιηΘ = ѕιηα тнєη,Θ = ηΠ (−1)^ηα
1. ѕιη2α = 2ѕιηα¢σѕα
2. ¢σѕ2α = ¢σѕ²α − ѕιη²α
3. ¢σѕ2α = 2¢σѕ²α − 1
4. ¢σѕ2α = 1 − ѕιη²α
5. 2ѕιη²α = 1 − ¢σѕ2α
6. 1 ѕιη2α = (ѕιηα ¢σѕα)²
7. 1 − ѕιη2α = (ѕιηα − ¢σѕα)²
8. тαη2α = 2тαηα / (1 − тαη²α)
9. ѕιη2α = 2тαηα / (1 тαη²α)
10. ¢σѕ2α = (1 − тαη²α) / (1 тαη²α)
11. 4ѕιη³α = 3ѕιηα − ѕιη3α
12. 4¢σѕ³α = 3¢σѕα ¢σѕ3α
» ѕιη²Θ ¢σѕ²Θ=1
» ѕє¢²Θ-тαη²Θ=1
» ¢σѕє¢²Θ-¢σт²Θ=1
» ѕιηΘ=1/¢σѕє¢Θ
» ¢σѕє¢Θ=1/ѕιηΘ
» ¢σѕΘ=1/ѕє¢Θ
» ѕє¢Θ=1/¢σѕΘ
» тαηΘ=1/¢σтΘ
» ¢σтΘ=1/тαηΘ
» тαηΘ=ѕιηΘ/¢σѕΘ continue...
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Wednesday, 2 August 2017
Maths formulla (rectangle) in hindi
आयत के फ़ॉर्मूलास : √ल ० का वर्ग + चौ ० का वर्ग
आयत का विकर्ण =
आयत का क्षेत्रफल = ल *चौ
आयत का परिमिति या परिमाप = २ (ल +चौ )
आयत के क्षेत्रफल में वृद्धि = a+b+ab/100
आयत के क्षेत्रफल में कमी = a-b-ab/100
एक आयताकार खेत की ल ० a और चौ b तथा इसके चारो ओर x चौ का रास्ता हो तो रस्ते का क्षेत्रफल = 2x (a+b+2x)
एक आयताकार खेत की ल ० a और चौ b तथा इसके अंदर चारो ओर x चौ का रास्ता हो तो रस्ते का क्षेत्रफल = 2x (a+b-2x)
Area
वर्ग का परिमाप = 4A
वर्ग का क्छेत्रफल = A *A
Area of square = a2
Dimension of square= 4a where a is side of square.
Circle
वृत के क्षेत्रफल = π*r2
cirumference of circle = 2πr
पाइथागोरस प्रमेय - किसी समकोण त्रिभुज में कर्ण का वर्ग अन्य दो भुजाओ के वर्गों के योग के वरावर होता है।
A2+B2=C2 where A and B आधार और लंब और C कर्ण है।
आयत का विकर्ण =
आयत का क्षेत्रफल = ल *चौ
आयत का परिमिति या परिमाप = २ (ल +चौ )
आयत के क्षेत्रफल में वृद्धि = a+b+ab/100
आयत के क्षेत्रफल में कमी = a-b-ab/100
एक आयताकार खेत की ल ० a और चौ b तथा इसके चारो ओर x चौ का रास्ता हो तो रस्ते का क्षेत्रफल = 2x (a+b+2x)
एक आयताकार खेत की ल ० a और चौ b तथा इसके अंदर चारो ओर x चौ का रास्ता हो तो रस्ते का क्षेत्रफल = 2x (a+b-2x)
Area
वर्ग का परिमाप = 4A
वर्ग का क्छेत्रफल = A *A
Area of square = a2
Dimension of square= 4a where a is side of square.
Circle
वृत के क्षेत्रफल = π*r2
वृत की परिधि = 2πr
Area of circle = πr2cirumference of circle = 2πr
पाइथागोरस प्रमेय - किसी समकोण त्रिभुज में कर्ण का वर्ग अन्य दो भुजाओ के वर्गों के योग के वरावर होता है।
A2+B2=C2 where A and B आधार और लंब और C कर्ण है।
Saturday, 29 July 2017
Zero (शून्य smoke)
शून्य (0) एक अंक है जो संख्याओं के निरूपण के लिये प्रयुक्त आजकी सभी स्थानीय मान पद्धतियों का अपरिहार्य प्रतीक है। इसके अलावा यह एक संख्या भी है। दोनों रूपों में गणित में इसकी अत्यन्त महत्वपूर्ण भूमिका है। पूर्णांकों तथा वास्तविक संख्याओं के लिये यह योग का तत्समक अवयव(additive identity) है।
- किसी भी वास्तविक संख्या को शून्य से गुणा करने से शून्य प्राप्त होता है। (x * 0 = 0)
- किसी भी वास्तविक संख्या को शून्य से जोड़ने या घटाने पर वापस वही संख्या प्राप्त होती है। (x + 0 = x ; x - 0 = x)
शून्य का आविष्कार किसने और कब किया यह आज तक अंधकार के गर्त में छुपा हुआ है, परंतु सम्पूर्ण विश्व में यह तथ्य स्थापित हो चुका है कि शून्य का आविष्कार भारत में ही हुआ। ऐसी भी कथाएँ प्रचलित हैं कि पहली बार शून्य का आविष्कार बाबिल में हुआ और दूसरी बार माया सभ्यता के लोगों ने इसका आविष्कार किया पर दोनो ही बार के आविष्कार संख्या प्रणाली को प्रभावित करने में असमर्थ रहे तथा विश्व के लोगों ने इन्हें भुला दिया। फिर भारत में हिंदुओंने तीसरी बार शून्य का आविष्कार किया। हिंदुओं ने शून्य के विषय में कैसे जाना यह आज भी अनुत्तरित प्रश्न है। अधिकतम विद्वानों का मत है कि पांचवीं शताब्दी के मध्य में शून्य का आविष्कार किया गया। सर्वनन्दि नामक दिगम्बर जैन मुनि द्वारा मूल रूप से प्रकृत में रचित लोकविभागनामक ग्रंथ में शून्य का उल्लेख सबसे पहले मिलता है। इस ग्रंथ में दशमलव संख्या पद्धति का भी उल्लेख है
अर्थात् "एक, दश, शत, सहस्र, अयुत, नियुत, प्रयुत, कोटि, अर्बुद तथा बृन्द में प्रत्येक पिछले स्थान वाले से अगले स्थान वाला दस गुना है।"[1] और शायद यही संख्या के दशमलव सिद्धान्त का उद्गम रहा होगा। आर्यभट्ट द्वारा रचित गणितीय खगोलशास्त्र ग्रंथ 'आर्यभट्टीय' के संख्या प्रणाली में शून्य तथा उसके लिये विशिष्ट संकेत सम्मिलित था (इसी कारण से उन्हें संख्याओं को शब्दों में प्रदर्शित करने के अवसर मिला)। प्रचीन बक्षाली पाण्डुलिपि में, जिसका कि सही काल अब तक निश्चित नहीं हो पाया है परन्तु निश्चित रूप से उसका काल आर्यभट्ट के काल से प्राचीन है, शून्य का प्रयोग किया गया है और उसके लिये उसमें संकेत भी निश्चित है। उपरोक्त उद्धरणों से स्पष्ट है कि भारत में शून्य का प्रयोग ब्रह्मगुप्त के काल से भी पूर्व के काल में होता था।
शून्य तथा संख्या के दशमलव के सिद्धान्त का सर्वप्रथम अस्पष्ट प्रयोग ब्रह्मगुप्त रचित ग्रंथ ब्राह्मस्फुटसिद्धान्त में पाया गया है। इस ग्रंथ में ऋणात्मक संख्याओं (negative numbers) और बीजगणितीय सिद्धान्तों का भी प्रयोग हुआ है। सातवीं शताब्दी, जो कि ब्रह्मगुप्त का काल था, शून्य से सम्बंधित विचार कम्बोडिया तक पहुँच चुके थे और दस्तावेजों से ज्ञात होता है कि बाद में ये कम्बोडिया से चीनतथा अन्य मुस्लिम संसार में फैल गये।
इस बार भारत में हिंदुओं के द्वारा आविष्कृत शून्य ने समस्त विश्व की संख्या प्रणाली को प्रभावित किया और संपूर्ण विश्व को जानकारी मिली। मध्य-पूर्व में स्थित अरब देशों ने भी शून्य को भारतीय विद्वानों से प्राप्त किया। अंततः बारहवीं शताब्दी में भारत का यह शून्य पश्चिम में यूरोप तक पहुँचा।
भारत का 'शून्य' अरब जगत में 'सिफर' (अर्थ - खाली) नाम से प्रचलित हुआ। फिर लैटिन, इटैलियन, फ्रेंच आदि से होते हुए इसे अंग्रेजी में 'जीरो' (zero) कहते हैं
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