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If S and T are two sets such that S has 21 elements, T has 32 elements, and
S ∩ T has 11 elements, how many elements does S ∪ T have?
n(S)=21
n(T)=32
n(S ∩ T)=11
n(S ∪ T)=n(T)+n(S)-n(S ∩ T)
=21+32-11=42
If X and Y are two sets such that X has 40 elements, X ∪Y has 60 elements and X ∩Y has 10 elements, how many elements does Y have?
n(X)= 40
n(XUY)= 60
n(X∩Y)= 10
n(XUY)= n(X) + n(Y) -n(X∩Y)
60= 40 + n(Y)- 10
n(Y) = 60 - 40 + 10 = 30
Thus, the set Y has 30 elements.
In a group of 70 people, 37 like coffee, 52 like tea, and each person likes at least one of the two drinks. How many people like both coffee and tea?
C-coffee T-tea
n(C)=37
n(T)=52
n(CUT)=70
n(CUT)=n(C)+n(T)-n(C∩T)
70=37+52-n(C∩T)
n(C∩T)=37+52-70=19
since each person likes at least one of the two drinks,n(U)=n(CUT)
number of peoples like both tea and coffee=19
If X and Y are two sets such that n(X) = 17, n(Y) = 23 and n(X ∪ Y) = 38, find n(X ∩Y).
It is given that:
n(X) = 17, n(Y) = 23, n(X ∪ Y) = 38
n(X ∩ Y) = ?
We know that:
If X and Y are two sets such that X ∪Y has 18 elements, X has 8 elements and Y has 15 elements; how many elements does X ∩Y have?
It is given that:
n(X ∩ Y) = ?
We know that:
In a group of 400 people, 250 can speak Hindi and 200 can speak English. How many people can speak both Hindi and English?
number of people speak hindi, n(H)=250
number of people speak english, n(E)=200
n (HUE) = 400
n(HUE)= n(H) +n(E) -n(H∩E)
400 = 250+200 - n(H∩E)
=450-n(H∩E)
n(H∩E)=450-400= 50
Thus, 50 people can speak both Hindi and English.
In a group of 65 people, 40 like cricket, 10 like both cricket and tennis. How many like tennis only and not cricket? How many like tennis?
Let C denote the set of people who like cricket, and
T denote the set of people who like tennis
∴ n(C ∪ T) = 65, n(C) = 40, n(C ∩ T) = 10
We know that:
n(C ∪ T) = n(C) + n(T) – n(C ∩ T)
∴ 65 = 40 + n(T) – 10
⇒ 65 = 30 + n(T)
⇒ n(T) = 65 – 30 = 35
Therefore, 35 people like tennis.
Now,
(T – C) ∪ (T ∩ C) = T
Also,
(T – C) ∩ (T ∩ C) = Φ
∴ n (T) = n (T – C) + n (T ∩ C)
⇒ 35 = n (T – C) + 10
⇒ n (T – C) = 35 – 10 = 25
Thus, 25 people like only tennis.
In a committee, 50 people speak French, 20 speak Spanish and 10 speak both Spanish and French. How many speak at least one of these two languages?
Let F be the set of people in the committee who speak French, and
S be the set of people in the committee who speak Spanish
∴ n(F) = 50, n(S) = 20, n(S ∩ F) = 10
We know that:
n(S ∪ F) = n(S) + n(F) – n(S ∩ F)
= 20 + 50 – 10
= 70 – 10 = 60
Thus, 60 people in the committee speak at least one of the two languages.
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