5. Let X be the set X = {1,2,3}, and let R be the relation defined on X by x₁ Rx₂ if x1 + x2 3. (a) Write R as a subset of X X X. (b) Determine if R is reflexive, symmetric, antisymmetric, and/or transitive.

For the set A = {are, bot, cab, era}, define the relation R on A to be: (x,y) e R if and only if word x and word y share one letter only. For example (are, cab) is in R because those words share one letter only ("a"). But (are, era) is NOT in R because those words do NOT share one letter only, they share 3 letters. Also (bot, era) is NOT in R since they share no letters at all. In cach part, answer yes or no. If yes, then explain why in terms of this specific relation about sharing letters (not just by stating the generic definition of the property). If you say a) Is R reflexive? b) Is R symmetric? c) Is R antisymmetric? d) Is R transitive?

Let R be the relation on R defined by Ry if and only if ry 1. Thus R can also be represented as (r, y) ry = 1} . Use a similar representation for each of your answers to the questions below, and write your answer in the accompanying box What is the composite relation R2 ? (a) (b) What is the composite relation R3? What is the composite relation R4? (c) (d) What is the transitive closure of R?

Let R be the relation on the set of ordered pairs of positive integers such that ((a,b), (c,d)) e R if and only if a + d = b + c. Determine whether R is reflexive, symmetric and transitive.