Math Problem Solver

### (a) Prove that \( A \cap (B \cup C) = (A \cap B) \cup (A \cap C) \)

This is a distributive property of sets. You can prove it using element arguments or Venn diagrams.

### (b) Define \( A - B \), the complement of \( B \) in \( A \)

\( A - B \) is the set of elements that are in \( A \) but not in \( B \).

#### (i) Use the algebra of sets to show that:

\( (A - C) = (A - B) - (C - B) \)

You can use set identities and properties to prove this.

### (c) Let \( A = \{1, 2, 3\} \)

#### (i) List the elements of the power set \( P(A) \)

The power set \( P(A) \) includes all subsets of \( A \).

#### (ii) Define what is meant by a partition of \( P(A) \)

A partition of \( P(A) \) is a division of \( P(A) \) into non-overlapping subsets.

#### (iii)

1. Give two distinct examples of partitions of \( P(A) \).

2. Decide whether the given statements are true or false.