Question

The following definitions are used: a relation on a set A is defined to be irreflexive if, and only if, for every x A, x R x; asymmetric if, and only if, for every x, y A if x R y then y R x; intransitive if, and only if, for every x, y, z A, if x R y and y R z then x R z. The following relation is defined on the set A = {0, 1, 2, 3}. Determine whether the relation is irreflexive, asymmetric, intransitive, or none of these. (Select all that apply.) R2 = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)}

Answer 1

The relation R2 = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)} is not irreflexive, asymmetric, or intransitive.

Step-by-step explanation:

To determine whether the relation R2 = {(0, 0), (0, 1), (1, 1), (1, 2), (2, 2), (2, 3)} is irreflexive, asymmetric, or intransitive, we will analyze each property.

1. Irreflexive: Since every element in the relation has the form (x, x) where x is an element of A, and all such pairs exist in R2, the relation is not irreflexive.
2. Asymmetric: Since the relation contains (x, y) and (y, x) pairs for some x, y in A, it violates the definition of an asymmetric relation. Therefore, R2 is not asymmetric.
3. Intransitive: The relation R2 does not violate the transitive property. For example, (0,1) and (1,2) are in R2, and it also contains (0,2), satisfying the transitive property. Therefore, R2 is not intransitive.

Therefore, the relation R2 is none of the given properties (irreflexive, asymmetric, or intransitive).