Let A = {1, 2, 3, 4}. Define a relation R on A by a R b ⇐⇒ a + b ≤ 4 for every a, b ∈ A. (a) List all the elements of R. (b) Determine whether R has the following properties. If R has a certain property, prove this is so, otherwise, provide a counterexample to show that it does not. i. Reflexivity ii. Transitivity iii. Antisymmetry iv. Symmetry

(a) { (1,1), (1,2), (2,1), (1,3), (3,1), (2,2) }

(b)

i. The relation is not reflexive, because (3,3) and (4,4) do not belong to R

ii. The relation is not transitive, because (2,1) and (1,3) belong to R, but (2,3) does not belong to R

iii. The relation is not antisymmetric, because if a + b ≤ 4, then b + a ≤ 4. And we can see that (1,3) and (3,1) belong to R, but 1 is not equal to 3.

iv. If a + b ≤ 4, then b + a ≤ 4, so the relation is symmetric.