Question

8) Let R be a relation on the set A = {a, b, c} defined by R= {(a, a),(a, b), (a, c), (b, a), (b, b)}. (3 points)_a) Find Mr, the zero-one matrix representing R (with the elements of the set listed in alphabetical order). (2 points)_b) Is R reflexive? If not, give a counterexample. (2 points)_c) Find the symmetric closure of R. (3 points)_d) Find MR O MR.

Answer 1

a) M r : Matrix(m*n) Representation of a relation 1: if it belongs in R otherwise 0

b) R is not reflexive

c) R U R` = Symmetric Closure [R` : Inverse of Relation]

d)It means consider only entries having length of 2

Please PFA for clear explanation:

Relation relation m Sol A =şa,b,c}| $(0,0), (6,2),10,0),(a,b), (bb) RE {ta, a) , (a,b),(0,9 ,16,0),(6,5)} R be a defined by : R = R = {la,a) la,b),(a,c) (b,a), (b,b?? Mei myn matrix , hare man RUR → Symmetrie, clown on = {(0,0),(0,6), (0,0),(6,a), C,0) sk.by meg 1; if (a, b;) ER ; if ( a., b;) ¢ R = 0 a b C Here, d) MROMR Theorem : 9. R is a relatjon on A = {a,,2,,- an} MROR а. 1 1 1 mer b 1 1 0 (a, a),(a,b), (a,0),(b,b) 6, a) ER m MA C 0 16,0), (ca), (C,b), (C,C)&R MROMR = soo - С. 1 = 2 b O ob obo M.2 "R L 0 o b) під not Reflexive Reflexive Relation is one in which every element maps to itself. Giren R is not reflexive because (C,C) & R. so R would be reflexive if eg{2, a), (6,6), (C,0} bresent in set A= {a,b,c}. Now ne have consider only baths having distance/ length af. Ma come = { 0,02), (2,6),(,0),(6,6) 9 Symmetrie closure Diagonal pair : laa) ,(6,6) Non-Diagonal baix : (a,b),@,C){ba) For Non-Diagonal pairs we have take their inverse also ie (a,b), (6,a), la,c), (c)a) it R = {a,b)] then ' = {(6,7)} RUR' Symmetric Relatim to