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) Define the relation M(A B) A ∩ B 6= ∅ where the domains for A and B are all subsets of Z For each of

Define the relation M(A, B) : AnB # 0, where the domains for A and B are all subsets of
Z. For each of the five properties of a relation defined in this chapter (reflexive, irreflexive,
symmetric, antisymmetric, and transitive) either show M satisfies the property, or explain
why it does not.
Hint: This problem causes a lot of grief. The relation M is a relation between subsets of
Z. For example, {1, 2}M{9} is false, since {1, 2} n {9} = 0. But {1, 2} M {2, 4,6, 7} is true
since {1, 2} n {2, 4, 6, 7} = {2}, and not 0.
(bonus) Let A = 0 and consider the empty relation, E = 0, on A. For each of the five prop-
erties of a relation defined in this chapter (reflexive, irreflexive, symmetric, antisymmetric,
and transitive) either show M satisfies the property, or explain why it does not.
Warning: Reasoning about the empty set can cause grave mental anguish. Suggestion:
write out each of the five definitions you need to check (reflexive, symmetric, etc.) and
decide if the statement is true or false.
For example: For the reflexive condition, the statement to check is
Vr(c E0 = (1,I) ( E)
Since the left side of the implication (x 6 0) is definitely false (after all there isn’t any-
thing in the empty set!) the entire implication is true (recall the truth table for implication).
That means the proposition above is true, and so E is reflexive. Reasoning for the remain-
ing four conditions all follow a similar pattern.
Math