Let A = {1,2,3,4}. Define a relation R on A by a R b ↔ a + b ≤ 4 for every a; b ϵ 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

Given, $A=\{1,2,3,4\}$ and $R=\{(a,b):a+b \leq 4\}.$

a) The elements of the relation R is, $R=\{(1,1),(1,2),(1,3),(2,1),(2,2),(3,1)\}$.

b)

i) R is not reflexive, since $(3,3)\notin R$

ii) R is not transitive, since $(2,1)\in R, (1,3)\in R$, but $(2,3) \notin R$

iii) R is not anti-symmetry, since $(1,2)\in R, (2,1)\in R$, but $1\neq 2.$

iv) For each $(a,b)\in R$ , we have $(b,a)\in R$. Hence, R is symmetry.