Question

Suppose that \(R\) and \(S\) are reflexive relations on a set \(A\).

Prove or disprove each of these statements.

a) \(R \cup S\) is reflexive.

b) \(R \cap S\) is reflexive.

c) \(R \oplus S\) is irreflexive.

d) \(R - S\) is irreflexive.

e) \(S^\circ R\) is reflexive.

Short answer

(a)\(R \cup S\)is reflexive

(b)\(R \cap S\)is reflexive

(c)\(R \oplus S\)is irreflexive

(d)\(R - S\)is irreflexive

(e) \(S^\circ R\) is irreflexive

Step-by-step solution

Given Data

\(R\) and \(S\) are reflexive relations on a set \(A\)

Concept of the reflexive, irreflexive, union, intersection, difference, symmetric difference and composite

Reflexive: A relation \(R\) on a set \(A\) is called reflexive if \((a,a) \in R\) for every element \(a \in A\).

Irreflexive: A relation\(R\)on a set\(A\)is irreflexive if\((a,a) \notin R\)for every element\(a \in A\).

Union\(A \cup B\): All elements that are either in\(A\)OR in\(B\)

Intersection\(A \cap B\): All elements that are both in\(A\)AND in\(B\).

Difference\(A - B\): All elements in\(A\)that are NOT in\(B\)(complement of\(B\)with respect to\(A\)).

Symmetric difference\(A \oplus B\): All elements in\(A\)or in\(B\), but not in both.

The composite \(S^\circ R\) consists of all ordered pairs \((a,c)\) for which there exists an element \(b\)such that \((a,b) \in R\) and \((b,c) \in S\)

Determine the \(R \cup S\) is reflexive

(a)

Let \(a \in A\)

Since \(R\) is reflexive:

\((a,a) \in R\)

Since \(S\) is reflexive:

\((a,a) \in S\)

The union \(R \cup S\)contains all elements in either \(R\) or \(S\).

\((a,a) \in R \cup S\)

Be the definition of reflexive, \(R \cup S\) is then reflexive.

Determine the \(R \cap S\) is reflexive

(b)

Let \(a \in A\)

Since \(R\)is reflexive:

\((a,a) \in R\)

Since \(S\) is reflexive:

\((a,a) \in S\)

The union \(R \cap S\) contains all elements in both \(R\) and \(S\).

\((a,a) \in R \cap S\)

Be the definition of reflexive, \(R \cap S\) is then reflexive.

Determine the \(R \oplus S\) is irreflexive

(c)

Let \(a \in A\)

Since \(R\) is reflexive:

\((a,a) \in R\)

Since \(S\) is reflexive:

\((a,a) \in S\)

The symmetric difference \(R \oplus S\)contains all elements in \(R\) or \(S\), but not those that are in both sets.

\((a,a) \notin R \oplus S\)

Be the definition of reflexive, \(R \oplus S\) is then irreflexive.

Determine the \(R - S\) is irreflexive

(d)

Let \(a \in A\)

Since \(R\) is reflexive:

\((a,a) \in R\)

Since \(S\) is reflexive:

\((a,a) \in S\)

The difference \(R - S\)contains all elements in \(R\) that are not in \(S\) as well.

\((a,a) \notin R - S\)

Be the definition of reflexive, \(R - S\) is then irreflexive.

Determine the \(S^\circ R\)is reflexive

(e)

Let \(a \in A\)

Since \(R\) is reflexive:

\((a,a) \in R\)

Since \(S\) is reflexive:

\((a,a) \in S\)

By the definition of the composite, if \((a,a) \in R\)

and \((a,a) \in S\), then:

\((a,a) \in S^\circ R\)

Be the definition of reflexive, \(S^\circ R\)is then reflexive

Therefore, (a)\(R \cup S\)is reflexive

(b)\(R \cap S\)is reflexive

(c)\(R \oplus S\)is irreflexive

(d)\(R - S\)is irreflexive

(e) \(S^\circ R\) is irreflexive