Let A = {1, 2, 3, 4, 5, 6} and R = {(a, b): b = a + 2}. Write R in roster form and check whether R is reflexive. Write the smallest reflexive relation on A that contains R.

1. 1 · Write what is given and what is to be found. Given: A = {1, ..., 6}, R = {(a, b): b = a + 2}. To find: R in roster form, whether R is reflexive, and the smallest reflexive relation containing R.
2. 2 · Write R in roster form. R = {(1, 3), (2, 4), (3, 5), (4, 6)}
3. 3 · Write the rule. R is reflexive if (a, a) ∈ R for every a ∈ A.
4. 4 · Test the rule. (1, 1) ∉ R, because 1 ≠ 1 + 2. In fact no pair (a, a) is in R, since a ≠ a + 2. So R is not reflexive.
5. 5 · Add the missing pairs. R ∪ {(1, 1), (2, 2), (3, 3), (4, 4), (5, 5), (6, 6)}. This has 4 + 6 = 10 pairs.
6. 6 · Check and box the answer. R is not reflexive. The smallest reflexive relation containing R has 10 pairs: the 4 pairs of R and the 6 pairs (a, a).

Scoring tip: When A is small, write R in roster form first. It makes every check visible to the examiner.