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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 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (4,4)} on the set A = {1,2,3,4} by working as follows. (i) Find the reflexive closure R of U, R = reflexive (U) = UUAA=UU {(1, 1), (2, 2), (3, 3), (4,4)}. (ii) Find the symmetric closure S of R. S = symmetric (R) = RUR¹.

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 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (4,4)} on the set A = {1,2,3,4} by working as follows. (i) Find the reflexive closure R of U, R = reflexive (U) = UUAA=UU {(1, 1), (2, 2), (3, 3), (4,4)}. (ii) Find the symmetric closure S of R. S = symmetric (R) = RUR¹.

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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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 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (4,4)} on the set A = {1,2,3,4} by working as follows. (i) Find the reflexive closure R of U, R = reflexive (U) = UUAA = UU {(1, 1), (2, 2), (3, 3), (4,4)}. (ii) Find the symmetric closure S of R, S = symmetric (R) = RUR¹.
Transcribed Image Text: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 = {(1, 1), (1, 2), (2, 1), (2, 2), (3, 1), (4,4)} on the set A = {1,2,3,4} by working as follows. (i) Find the reflexive closure R of U, R = reflexive (U) = UUAA = UU {(1, 1), (2, 2), (3, 3), (4,4)}. (ii) Find the symmetric closure S of R, S = symmetric (R) = RUR¹.
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