Question

For non-empty sets A, B and C, the following two statements are given:
Statement `P:Ann(BuuC)=(AnnB)uuC`
Statement `Q : C` is a subset of A
Which one of the following is correct ?
A. `P le Q`
B. `P iff Q`
C. `P implies Q`
D. Nothing can be said about the correctness of the above three with certainty

Answer 1

Correct Answer - B
A Venn diagram is drawn for 3 intersecting set A, B, C under a universal set U, creating 8 regions in total named, a to h as shown

Statement `P:Ann(BuuC)=(AnnB)uuC`
`LHS-=a,b,c,d,nn(b,c,d,e,f,g,)-=b,c,d`.
`RHS-=(c,d)uu(b,c,f,g)=b,c,d,f,g`
If P is correct then region f, g do not lie in set C and set C has regions b, c only.
This follows that C is subset of A. Since, Set A has regions a, b, c, d and C has regions b, c.
Thus, `P iff Q`.
Also, if C is a subset of A, Q is true, then the Venn diagram appears as below:
`{Ann(BuuC)}`

LHS of P statement gives region a, b, c, d `nn` region b, c, d, e `-=` b, c, d.
`RHS: {(AnnB)uuC}` gives : region c, d, `uu` region b, c.
`-=b, c, d`
and LHS = RHS shows `Q iff P`,
Comparing both gives `P iff Q`.