PROBLEM & SOLUTION BY APPLYING VENN DIAGRAM FORMULA
Formula of Venn-Diagram -
1) n(A U B) + n(A ∩ B) = n(A) + n(B)
or, n(A U B) = n(A) + n(B) - n(A ∩ B)
2) n( A U B) = n (A) + n (B)
3) n(A) + n(A)’ = n(U)
4) n(A U B) = n (B) + n ( A – B )
5) n(A) = n(A ∩ B) + n( A – B )
6) n (A – B) + n ( B – A ) = n(A U B) - n(A ∩ B)
Example.-A) Let U be the universal set and P, Q be any two set , If n(U) = 100 , n(P) = 50 , n(Q) = 25 and n(P ∩ Q)’ = 40, find the following
1) n(P – Q) , 2) n(P ∩ Q) , 3) n ( P U Q ) , 4) n ( Q – P )
1) n(P – Q)
Ans.) n( P – Q ) + n(Q) = n ( P U Q )
=> n( P – Q ) = n ( P U Q ) - n(Q)
Now we have to find out the value of n ( P U Q )
So, n(P U Q) = n(P) + n(Q) – n (P ∩ Q) = 15
Now as per the given condition and requirement –
n( P – Q ) = n ( P U Q ) - n(Q)
so, n( P – Q ) = 15 – 25 = - 10 (Ans.)
2) n(P ∩ Q)
Answer) we have to find the value of n ( P ∩ Q )
As per the formula we know that n(A) + n(A’) = n(U)
Replacing the A by P ∩ Q, then we get –
n(P ∩ Q) + n(P ∩ Q)’ = n(U)
=> n(P ∩ Q) + 40 = 100
n(P ∩ Q) = 100 – 40 = 60 (Ans.)
3) n ( P U Q )
Answer) we have to find the value of n ( P U Q )
As per the formula we know that n(A U B) = n(A) + n(B) – n (A ∩ B)
Replacing A & B by P & Q, we get n(P U Q) = n(P) + n(Q) – n (P ∩ Q)
When n(P ∩ Q) + n(P ∩ Q)’ = n(U), then n(P ∩ Q) = 60
So, n(P U Q) = n(P) + n(Q) – n (P ∩ Q)
= 50 + 25 – 60 = 15 (Ans.)
4) n ( Q – P )
As per the formula we know that,
n(A – B ) + n(B – A) = n(A U B) – n(A ∩ B)
Now we will replace A & B with P & Q we get –
n(P – Q ) + n(Q – P) = n(P U Q) – n(P ∩ Q)
now, n(Q – P) = n(P U Q) – n(P ∩ Q) - n(P – Q )
= 15 – 60 – (-10)
= 15 – 60 + 10
= - 35 (Ans.)
Exercise-B) Let U be the universal set and A & B any two sets. If n(U) = 20, n(A) = 6, n(B) = 4 and n(A U B )’ = 8 , then find the following
1) n(A)’ , 2) n(B)’ , 3) n (AUB), 4) n (A ∩ B), 5) n(A – B ), 6) n (B – A )
1) n(A)’
Ans.) according to the formula, we get n(A) + n(A)’ = n(U)
Now, replacing the value of n(A) & n(U) we can get,
6 + n(A)’ = 20
Or, n(A)’ = 14 (Ans.)
2) n(B)’
Ans.) according to the formula, we get n(B) + n(B)’ = n(U)
Now, replacing the value of n(B) & n(U) we can get,
So, 4 + n(B)’ = 20
n(B)’ = 20 – 4 = 16 (Ans.)
3) n (AUB)
Ans.) according to the formula, we get n(A) + n(A)’ = n(U)
Now, replacing A by n(AUB)
n(AUB) + n(AUB)’ = n(U)
replacing the value of n(U) = 20 & n(A U B )’ = 8 we get,
n(AUB) + 8 = 20
now, n(AUB) = 20 – 8 = 12 (Ans.)
4) n (A ∩ B)
Ans.) according to the formula, we get
n(A U B) = n(A) + n(B) – n(A ∩ B)
First, we have to find the value of n(A U B)
Now, replacing A by n(AUB)
n(AUB) + n(AUB)’ = n(U)
replacing the value of n(U) = 20 & n(A U B )’ = 8 we get,
n(AUB) + 8 = 20
now, n(AUB) = 20 – 8 = 12
Now, n(A U B) = n(A) + n(B) – n(A ∩ B)
Replacing the value n(A U B) = 12 , n(A) = 6, n(B) = 4
12 = 6 + 4 - n(A ∩ B)
n(A ∩ B) = 12 – 10 = 2 (Ans.)
5) n ( A – B )
Ans.) According to the formula, we get n(A) – n(A – B) = n(A ∩ B)
we have to replace the value of n(A ∩ B) and n(A)
n(A) – n(A – B) = n(A ∩ B)
n(A) - n(A ∩ B) = n(A – B)
n(A – B) = n(A) - n(A ∩ B) = 6 – 2 = 4 (Ans.)
6) n (B – A )
Ans.) according to the formula, we get
n(A – B) + n(B – A ) = n( A U B ) - n(A ∩ B)
Now, n(B – A ) = n( A U B ) - n(A ∩ B) - n(A – B)
Replace the value of n( A U B ), n(A ∩ B) & n(A – B)
n(B – A ) = 12 – 2 – 4
= 6 (Ans.)
Example.-C) Let, A & B be any two sets. If n(A – B) = 30, n(A U B) = 70, n(A ∩ B) = 20 then find
1) n(B – A ), 2) n(A), 3) n(B)
1) n (B – A)
Answer) According to the formula
n(A – B) + n(B – A) = n(A U B) – n(A ∩ B)
Now, replace the value of n(A – B) = 30, n( A U B ) = 50, n(A ∩ B) = 20
We get, n(A – B) + n(B – A) = n(A U B) – n(A ∩ B)
=> 30 + n(B – A) = 70 – 20
=> n(B – A) = 50 – 30 = 20 (Ans.)
2) n(A)
Answer) According to the formula
n(A) - n(A – B) = n(A ∩ B)
Now, replace the value of n(A – B) = 30, n( A U B ) = 50, n(A ∩ B) = 20
n(A) - n(A – B) = n(A ∩ B)
=> n(A) = n(A – B) + n(A ∩ B)
= 30 + 20 = 50 (Ans.)
3) n(B)
Answer) According to the formula
n(A – B) + n(B) = n(A U B)
Now, replace the value of n(A – B) = 30, n( A U B ) = 50, n(A ∩ B) = 20
n(A – B) + n(B) = n(A U B)
=> n(B) = n(A U B) - n(A – B)
= 50 - 30 = 20 (Ans.)