(c) Let A and B be two sets. Determine which of the following statements are true and which are false. Prove each statement that is true and give a counter-example for each statement that is false. i. P(AՈB) = P(A)ՈP(B) ii. P(AՍB) = P(A)UP(B) iii. P(A) U P(B) ⊆ CP(AUB)

Sets are groups of well-defined objects or components in mathematics. A set is denoted by a capital letter, and the cardinal number of a set is enclosed in a curly bracket to indicate how many members there are in a finite set. For example, set A is a collection of all the natural numbers, such as A = {1,2,3,4,5,6,7,8,…..∞}.

i) Let A ={0,1} and B ={1,2}

Thus, A Ո B = {1}

P(A Ո B) = {∅, {0}, {1}, {2}, {0,1} , {1,2}, {0,2}}

P(A) = {∅, {0}, {1}, {0,1}}

P(B) = {∅, {1}, {2},{1,2}}

P(A) Ո P(B) = {∅, {1}}

∴ P(A Ո B) ≠ P(A) Ո P(B)

So, the given statement is false.

ii) Let A ={0,1} and B ={1,2}

Thus, A Ս B = {0,1,2}

P(A Ս B) = {∅, {0}, {1}, {2}, {0,1} , {1,2}, {0,2}, {0,1,2}}

P(A) = {∅, {0}, {1}, {0,1}}

P(B) = {∅, {1}, {2},{1,2}}

P(A) U P(B) = {∅, {0}, {1}, {2}, {0,1} , {1,2}}

∴ P(A Ս B) ≠ P(A) U P(B)

So, the given statement is false.

iii)

If X⊆Y then P(X)⊆P(Y) (1)

If X⊆Z and Y⊆Z then X∪Y⊆Z. (2)

Necessarily A⊆A∪B,

P(A)⊆P(A∪B); (a)

similarly, B⊆A∪B,

P(B)⊆P(A∪B). (b)

From (a) and (b), using (2), we conclude:

P(A) ∪ P(B) ⊆ P(A ∪ B).

So, the given statement is true.

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