Discrete Mathematics University of Kentucky CS 275 Spring ... - MGNet

Examples: Let A = {1, 2, 3, 4}.
• R 1 = {(1,1), (1,2), (2,1), (2,2), (3,4), (4,1), (4,4)} is
o just a relation
• R 2 = {(1,1), (1,2), (2,1)} is
o symmetric
• R 3 = {(1,1), (1,2), (1,4), (2,1), (2,2), (3,3), (4,1), (4,4)} is
o reflexive and symmetric
• R 4 = {(2,1), (3,1), (3,2), (4,1), (4,2), (4,3)} is
o antisymmetric and transitive
• R 5 = {(1,1), (1,2), (1,3), (1,4), (2,1), (2,2), (2,3), (2,4), (3,3), (3,4), (4,1),
(4,4)} is
o reflexive, antisymmetric, and transitive
• R 6 = {(3,4)} is
o antisymmetric
Note: We will come back to these examples when we get around to
representations of relations that work in a computer.
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Note: We can combine two or more relations to get another relation. We use
standard set operations (e.g., 3, 2, #, 4, …).
Definition: Let R be a relation on a set A to B and S a relation on B to a set C.
Then the composite of R and S is the relation S!R such that if (a,b),R and
(b,c),S, then (a,c), S!R, where a,A, b,B, and c,C.
Definition: Let R be a relation on a set A. Then R n is defined recursively: R 1 = R
and R n =R n!1 !R , n>1.
Theorem: The relation R is transitive if and only if R0R n , n