Let S be a set that contains at least two different elements

Let S be a set that contains at least two different elements. Let R be the relation on P(S), the power set of S, defined by (X, Y) elementof R if and only if X intersection Y = emptyset. Determine whether R is reflexive, symmetric, antisymmetric, or transitive. Remember to justify your answers.

Solution

Reflexive - Relation is said to be reflexive if (a,a) belongs to the Relation

In this case, X (int) X won\'t be equal to null set, hence the relation is not reflexive

Symmetric - Relation is said to be symmetric if (a,b) belongs to R, then (b,a) must also belong to R

So we have X (int) Y = null set

Similarly Y (int) X will also be a nullset

Hence the relation is a symmetric relation

AntiSymmetric - The relation is not antisymmetrix

Transitive Relation - Relation is said to be transitive if (a,b) and (b,c) belongs to R, then (a,c) must also belong to R

From first we get

X (int) Y = null set

Y (int) Z = null set

X (int) Z won\'t necessary be a null set

Example X = {1,2,3,5}, Y = {4,6,7}, Z = {1,2,3}

X (int) Y = null

Y (int) Z = null

X (int) Z = {1,2,3}

Hence the relation is not reflexive, symmetric, not antisymmetric and not transitive