91Ó°ÊÓ

Let \(a, b, c, d, m \in \mathbf{Z}\) with \(m \geq 2 .\) Prove each. Let \(r\) be the remainder when \(a\) is divided by \(m .\) Then \(a \equiv r(\bmod m)\)

Let \(R\) be any relation on a set \(A .\) Prove each. \(R \cup R^{-1}\) is the smallest symmetric relation that contains \(R\) (Hint: Suppose there is a symmetric relation \(S\) such that \(R \subseteq S \subseteq\) \(R \cup R^{-1} . )\)

Determine if each relation from \(\\{a, b, c, d\\}\) to {0,1,2,3,4} is a function. $$\\{(a, 1),(b, 2),(c, 3)\\}$$

Let \(R\) and \(S\) be relations on a set. Prove each. \(R\) is symmetric if and only if \(R^{-1}\) is symmetric.

Let \(A\) and \(B\) be finite sets with \(|A|=m\) and \(|B|=n .\) Find the number of binary relations that can be defined: From \(A\) to \(B\)

Let \(a, b, c, d, m \in \mathbf{Z}\) with \(m \geq 2 .\) Prove each. If \(a \equiv r(\bmod m)\) and \(0 \leq r

Let \(a, b, c, d, m \in \mathbf{Z}\) with \(m \geq 2 .\) Prove each. Let \(r_{1}\) and \(r_{2}\) be the remainders when \(a\) and \(b\) are divided by \(m\) respectively. Then \(a \equiv b(\bmod m)\) if and only if \(r_{1} \equiv r_{2}(\bmod m).\)

Let \(R\) and \(S\) be relations on a set. Prove each. \(R\) is symmetric if and only if \(R^{-1}=R\)

Let \(R\) and \(S\) be relations on a set. Prove each. If \(R\) and \(S\) are symmetric, \(R \cup S\) is symmetric.

A relation \(R\) on the set \(\\{1,2, \ldots, n\\}\) is given in terms of its elements. Write an algorithm to find its adjacency matrix \(A .\)