Practical Problems Related to Sets

Friday, March 01, 2019

Hello friends, इस article में आज हम सीखेंगे की Sets से related Practical problems को कैसे solve करते हैं!

ये सीखने से पहले की - Sets से related Practical problems को कैसे solve करते हैं; हम कुछ formulae देख लेते हैं जिनका इस्तेमाल हम Sets से related Practical problems को solve करने के लिए करेंगे...

Important Formulae for Solving Practical Problems Related to Sets

Let A, B and C are finite sets,

(i) n(A') = n(U) - n(A)

(ii) n(A⋃B') = n(A) - n(A⋂B)

(iii) n(A⋃B) = n(A) + n(B) - n(A⋂B)

(iv) n(A⋂B) = n(A) + n(B) - n(A⋃B)

(v) n(A⋃B⋃C) = n(A) + n(B) + n(C) - n(A⋂B) - n(B⋂C) - n(A⋂C) + n(A⋂B⋂C)

(vi) n(A⋂B⋂C) = n(A⋃B⋃C) - n(A) - n(B) - n(C) + n(A⋂B) + n(B⋂C) + n(A⋂C)

तो ये थे कुछ Sets से related formulae जिनका इस्तेमाल हम Sets से related problem को solve करने की लिए करेंगे।

चलिए अब Sets से related कुछ problems देख लेते है।

Practical Problems Related to Sets

Ex.1. If X and Y are two sets such that X⋃Y has 50 elements, X has 28 elements and Y has 32 elements, how many elements does X⋂Y have?

Solution: n(X⋃Y) = 50
n(X) = 28
n(Y) = 32
Using formula,
n(X⋂Y) = n(X) + n(Y) - n(X⋃Y)
= 28 + 32 - 50
= 60 - 50
= 10

Ex.2. In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?

Solution: Let H be the group of 250 people who can speak Hindi and E be the group of 200 people who can speak English. So, H⋃E will be the group of all the 400 people. ( Here H⋃E is working as universal set).

So, n(U) = 400
n(H) = 250
n(E) = 200
Using formula,
n(H⋂E) = n(H) + n(E) - n(H⋃E)
= 250 + 200 - 400
= 450 - 400
= 50

Ex.3. In a team of Basketball of a school there are 21 boys, in Hockey team there are 26 boys and in Football team there are 29 boys. Now if 14 boys play Hockey and basketball, 15 boys play Hockey and Football, 12 boys play Football and Basketball and 8 boys play all the three games- Hockey, Football and Basketball. Find the total number of boys that are playing all the three games- Hockey, Football and Basketball.

Solution: Let B, H and F be the set of boys that are playing Basketball, Hockey and Football respectively.

So, n(B) = 21
n(H) = 26
n(F) = 29
n(H⋂B) = 14
n(H⋂F) = 15
n(F⋂B) = 12
n(B⋂H⋂F) = 8
Using formula,
n(B⋃H⋃F) = n(B) + n(H) + n(F) - n(B⋂H) - n(H⋂F) - n(F⋂B) + n(B⋂H⋂F)
= 21 + 26 + 29 - 14 - 15 - 12 + 8
= 84 - 41
= 43
So, the total number of boys that are playing all the three games- Hockey, Football and Basketball is 43.

Ex.4. In a group of 20 peoples, there are 8 peoples who take tea but not takes coffee and there are 13 peoples who take tea. How many peoples are there who take coffee but not take tea?

Solution: Let C be the set of peoples who take coffee and T be the set of peoples who take tea. So, C⋃T will be the set of all 20 peoples. (Here C⋃T is working as universal set).

So, n(C⋃T) = 20
n(T) = 13
n(T - C) = 8
So, n(C) = n(C⋃T) - n(T - C)
= 20 - 8
= 12
and, n(C⋂T) = n(T) - n(T - C)
= 13 - 8
= 5
So, n(C - T) = n(C) - n(C⋂T)
= 12 - 5
= 7
So, the number of peoples who take coffee but not take tea are 7.

Ex.5. In a group of 1000 peoples, 750 peoples talk in Hindi language and 400 peoples talk in Bengali language. How many peoples talk only in Hindi and how many peoples talk only in Bengali? Also tell the number of peoples who talk in both languages- Hindi and Bengali?

Solution: Let H be the set of peoples who talk in Hindi and B be the set of peoples who talk in Bengali. So, H⋃B will be the set of all 1000 peoples. (Here H⋃B is working as universal set).

So, n(H⋃B) = 1000
n(H) = 750
n(B) = 400
So, n(H - B) = n(H⋃B) - n(B)
= 1000 - 400
= 600
So, the number of peoples who talk in Hindi only is 600.

And, n(B - H) = n(H⋃B) - n(H)
= 1000 - 750
= 250
So, the number of peoples who talk in Bengali only is 250.

So, n(H⋂B) = n(H⋃B) - [n(H - B) + n(B - H)]
= 1000 - [600 + 250]
= 1000 - 850
= 150
So, the number of peoples who talk in both languages- Hindi and Bengali are 150.

Ex.6. In a survey it was found that 21 people liked product E, 26 liked product H and 29 liked product L. If 14 people liked products E and H, 12 people liked products H and L, 14 people liked products E and L and 8 liked all the three products. Find how many liked product L only.

Solution: Let E, H and L be the set of peoples that like product E, H and L respectively.

So, n(E) = 21
n(H) = 26
n(L) = 29
n(E⋂H) = 14
n(H⋂L) = 12
n(E⋂L) = 14
n(E⋂H⋂L) = 8

Let's draw the Venn Diagram of the problem,

It is clear from Venn Diagram that the number of people who like product L only is,
= 29 - (6 + 8 + 4)
= 29 - 18
= 11

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