For each of the following relations, determine whether they are reflexive, symmetric, anti- symmetric, and/or transitive, and give a brief justification for each property. a) R ⊆ Z × Z where xRy iff x = y 2 b) The empty relation: R ⊆ A × A, where A is a non-empty set and R = ∅.
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For each of the following relations, let us determine whether they are reflexive, symmetric, anti- symmetric, and/or transitive.
a) where iff
Since we conclude that the pair and hence is not a reflexive relation.
Since we conclude On the other hand, and hence It follows that the relation is not symmetric.
If and then and Therefore, and hence or If then If then It follows that If and
then or and hence the relation is antisymmetric.
Taking into account that and but we conclude that the relation is not transitive.
b) The empty relation: , where is a non-empty set and .
Let Since we conclude that the relation is not reflexive.
Since the statement is false, the implication is true, and hence this relation is symmetric.
Taking into account that the statement is false, the implication is true, and hence this relation is antisymmetric.
Since the statement is false, the implication is true, and hence this relation is transitive.