Let A = {0, 1, 2, 3}, and define a relation R₂ on A as follows: R₂ = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)}. (a) Which of the following is the directed graph of R₂? 0 (b) Is R2 reflexive? Justify your answer. Yes because when m = any integer ✓ then (m, m) E (c) Is R₂ symmetric? Justify your answer. Yes because 1, 2 V (1, R2 and ? ✓ 1 ? ✓ R2. (d) Is R2 transitive? Justify your answer. No because (0, 3 ✓ EV R2 and (1,?) ? R2, and ' ?v 2 ? R2. © } (N) Co

For any relation on the nonempty set, the inverse of is the relation defined by if and only if . Prove the following statements. is symmetric if and only if . is antisymmetric if and only if is a subset of . is asymmetric if and only if .

Let and be lines in a plane. Decide in each case whether or not is an equivalence relation, and justify your decisions. if and only ifand are parallel. if and only ifand are perpendicular.

A relation R on a nonempty set A is called asymmetric if, for x and y in A, xRy implies yRx. Which of the relations in Exercise 2 areasymmetric? In each of the following parts, a relation R is defined on the set of all integers. Determine in each case whether or not R is reflexive, symmetric, or transitive. Justify your answers. a. xRy if and only if x=2y. b. xRy if and only if x=y. c. xRy if and only if y=xk for some k in . d. xRy if and only if xy. e. xRy if and only if xy. f. xRy if and only if x=|y|. g. xRy if and only if |x||y+1|. h. xRy if and only if xy i. xRy if and only if xy j. xRy if and only if |xy|=1. k. xRy if and only if |xy|1.