Question

Consider the following set theoretic expressions X, Y and Z where U-union, ∩ -intersection and superscript ‘C’ is set complement :

X = P ∩ Q ∩ R

Y = (P ∩ Q) ∪ (P ∩ R) ∪ (Q ∩ R^C) 

Z = (Q ∪ P^C) ∩ (P ∪ R) ∪ (P ∪ Q) ∩ P ∩ (P^C ∪ R)

Then which of the following is true?

1. No two expressions represent same set
2. All three expressions represent same set
3. Only X and Z represent same set
4. Only Y and Z represent same set

Answer 1

Correct Answer - Option 1 : No two expressions represent same set

Given:

X = P ∩ Q ∩ R

Y = (P ∩ Q) ∪ (P ∩ R) ∪ (Q ∩ RC) 

Z = (Q ∪ PC) ∩ (P ∪ R) ∪ (P ∪ Q) ∩ P ∩ (PC ∪ R)

Formula used:

Property-1:  A ∪ B = B ∪ A

Property-2:  A ∩ B = B ∩ A

Property-3: A ∩ A^c = ϕ 

Property-4: A ∪ Ac = ∪  

Property-5:  (A ∪ B) ∩ (B ∪ C) = A ∪ (B ∩ C)

Property-6:  (A ∩ B) ∪ (B ∩ C) = A ∩ (B ∪ C)

Property-7: A ∪ ϕ  = ϕ ∪ A = A

Property-8: A ∩ ϕ  = ϕ ∩ A = A

Calculation:

X = P ∩ Q ∩ R        ........(1)

Y = (P ∩ Q) ∪ (P ∩ R) ∪ (Q ∩ RC) 

using property- 6

Y = P ∩ (Q ∪ R) ∪ (Q ∩ RC)        ......(2)

Z = (Q ∪ PC) ∩ (P ∪ R) ∪ (P ∪ Q) ∩ P ∩ (PC ∪ R)

⇒ Z = (Q ∪ PC) ∩ (P ∪ R) ∪ (P ∪ Q) ∩ (P ∩ PC) ∪ ( P ∩ R)       (using P-6)

⇒ Z = (Q ∪ PC) ∩ (P ∪ R) ∪ (P ∪ Q) ∩ ϕ ∪ ( P ∩ R)          (using P-3)

⇒ Z = (Q ∪ PC) ∩ (P ∪ Q) ∪ (P ∪ R) ∩ ( P ∩ R)    (using P-1, P-7)

⇒ Z = Q ∪ (P^c ∩ p) ∪ (P ∪ R) ∩ ( P ∩ R)     (using P-5)

⇒ Z = Q ∪ ϕ ∪ (P ∪ R) ∩ ( P ∩ R)     (using P-3)

⇒ Z = Q ∪ (P ∪ R) ∩ ( P ∩ R)                ......(3)

Hence, from equation (1), (2) and (3), we can conclude that no two expressions represent same set.