Q1 R

1. (Equivalence Relations). Recall that given a relation U on a set A, the transitive closure of the symmetric closure of the reflexive closure of U is the least equivalence E relation on A which contains U; the relation E is called the equivalence closure of the relation U. Use this result to find the equivalence closure of the relation U={(2,2),(2,3),(2,4),(3,3)} on the set A={1,2,3,4} by working as follows. (i) Find the reflexive closure R of U, R=reflexive(U)=UA=U{(1,1),(2,2),(3,3),(4,4)}. (ii) Find the symmetric closure S of R. S=symmetric(R)=RR1 (iii) Find the transitive closure T of the relat ion S, T=transitive(S)=SS2S3S4 (to find the relat ion T work as youldid in HW 4). (iv) By (i) and (iii), the relation T must be reflexive and transitive. Verify direotly that T is symmetric (as it must be since any power of a simmetric relation is also symmetric, why? , thereby proving that T is an rauivalence relation, and then find the equivalence