91Ó°ÊÓ

A relation \(R\) on a set \(A\) is irreflexive if no element of \(A\) is related to itself, that is, if \((a, a) \notin R\) for every \(a \in A .\) Determine if each relation is irreflexive. The less-than relation on \(\mathbb{R}\).

Let \(A, B,\) and \(C\) be any \(n \times n\) boolean matrices. Prove each. $$A \vee(B \vee C)=(A \vee B) \vee C$$

Let \(A, B,\) and \(C\) be any \(n \times n\) boolean matrices. Prove each. $$A \wedge(B \wedge C)=(A \wedge B) \wedge C$$

Let \(R\) and \(S\) be relations from \(A\) to \(B\). Prove each. $$\left(R^{-1}\right)^{-1}=R$$

Let \(R\) and \(S\) be relations from \(A\) to \(B\). Prove each. $$\text { If } R \subseteq S, \text { then } S^{\prime} \subseteq R^{\prime}$$

Let \(A, B,\) and \(C\) be any \(n \times n\) boolean matrices. Prove each. $$A \vee(B \wedge C)=(A \vee B) \wedge(A \vee C)$$

A relation \(R\) on a set \(A\) is irreflexive if no element of \(A\) is related to itself, that is, if \((a, a) \notin R\) for every \(a \in A .\) Determine if each relation is irreflexive. The relation \(i s\) a parent of on the set of people.

Let \(R\) and \(S\) be relations from \(A\) to \(B\). Prove each. $$\text { If } R \subseteq S, \text { then } R^{-1} \subseteq S^{-1}$$

Let \(A, B,\) and \(C\) be any \(n \times n\) boolean matrices. Prove each. $$A \wedge(B \vee C)=(A \wedge B) \vee(A \wedge C)$$

Determine if each relation on \(\\{a, b, c\\}\) is irreflexive. $$\\{(b, a),(c, a)\\}$$