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

Determine if the relation R = {(1,3), (1,4), (2,3), (2,4), (3,4)} is reflexive, symmetric, antisymmetric, or transitive.

For the relation R closure of R. {(1,2), (1,4), (3,3), (4, 1)} find (i) R2 (ii) R3 and (iii) transitive

Define a relation R₂ on the set A = {0, 1, 2, 3) 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? DO (b) Is R₂ reflexive, symmetric, transitive, or none of these? (Select all that apply.) R₂ is reflexive. R₂ is symmetric. R₂ is transitive. OR₂ is neither reflexive, symmetric, nor transitive. x

Find the transitive closure of the relation R={(1,2),(2,2), (2,3),(3,3)} on the set A={1,2,3}. Select one: O a. ={(2,2),(3,1),(2.2).(2,3).(3,3)}. O b. ={(1,1),(3,1),(2.2),(2,3),(3,3)}. O c. =((1,3),(3,1).(2,2), (2,3),(3,3)}. O d. ={(1,2),(1,3).(2,2),(2,3),(3,3)}. O e. ={(1,2),(3,1),(2,2),(2,3),(3,3)}.

11./ a) b) c) 11 1 ) Consider the relation on A={1,2,3) represented by the matrix 0 10 1 Determine if the relation is reflexive. If it is not reflexive, state why. Determine if the relation is symmetric. If it is not symmetric, state why. Sketch a digraph to represent the relation.

4. Let R be a relation on the set {1, 2,3, 4} such that R = {(1, 1), (1, 3), (2, 2), (2,3), (3,2), (3, 4)}. Represent R using a matrix and a directed graph. 5. For each of the following relations, indicate whether the relation is reflexive, anti-reflexive, symmetric, anti-symmetric and transitive. Justify your answers. а. R is a relation on Z such that (x, y) ER if and only if x + y = 0. b. R is a relation on the set of all people such that (a, b) e R if and only if a was born on the same day as b.

Consider the relation R = {(1, 1), (1,3), (2,4), (3, 1), (3,3), (4, 3),(4,5)} on the set A = {1,2,3,4,5,6}. Find the Symmetric and reflexive closure.

[1 1 1 01 1 c) The matrix M = represents the relations R defined on L1 1. the set A = {1, 2, 3, 4}. i) List the ordered pairs in the relation R. [2 ii) Draw the graph representing the relation R. [2 d) How many edges does a 50-regular graph with 100 vertices have? [2

Define a relation R- on the set A = {0, 1, 2, 3} as follows. R5 = {(0, 0), (0, 1), (0, 2), (1, 2)} (a) Which of the following is the directed graph of R,? (b) Is R, reflexive, symmetric, transitive, or none of these? (Select all that apply.) O R, is reflexive. O Rg is symmetric. R5 is transitive. 10 O R, is neither reflexive, symmetric, nor transitive.