Verify a distributive property of the Cartesian product, such as A × (B ∪ C) = (A × B) ∪ (A × C), for given sets.

How to lay out the answer. Follow the frame, then write it in your own words.

1 · Write what is given and what is to be found. List A, B and C in roster form. Write the identity you must verify.
2 · Write the formula or theorem. Definition: A × B = {(a, b) | a ∈ A, b ∈ B}.
3 · Work out the left side.
- Find B ∪ C first.
- Pair each element of A with each element of B ∪ C.
4 · Work out the right side.
- Write A × B and A × C in full.
- Take their union, listing each pair only once.
5 · Compare the two sides. Show that both sides have exactly the same pairs, and count them.
6 · Check and box the answer. Write 'LHS = RHS. Hence verified.' and box it.

Scoring tip: Write every pair with brackets and a comma. A missing pair costs marks, so count: the left side must have n(A) × n(B ∪ C) pairs.