91Ó°ÊÓ

Define a binary relation \(P\) on \(\mathbf{Z}\) as follows: For all \(m, n \in \mathbf{Z}\), \(m P n \Leftrightarrow m\) and \(n\) have a common prime factor. a. Is \(15 P 25\) ? b. \(22 P 27\) ? c. Is \(0 P 5\) ? d. Is \(8 P 8 ?\)

Define a relation \(R\) on the set of all real numbers \(\mathbf{R}\) as follows: For all \(x, y \in \mathbf{R}\), $$ x R y \quad \Leftrightarrow \quad x^{2} \leq y^{2} . $$ Is \(R\) a partial order relation? Prove or give a counterexample.

A is the set of all strings of length 4 in \(a\) 's and \(b\) 's. \(R\) is defined on \(A\) as follows: For all strings \(s\) and \(t\) in \(A\), \(s R t \Leftrightarrow\) the first two characters of \(s\) equal the first two characters of \(t\).

Determine whether the given binary relation is reflexive, symmetric, transitive, or none of these. Justify your answers. \(R\) is the "greater than or equal to" relation on the set of real numbers: For all \(x, y \in \mathbf{R}, x R y \Leftrightarrow x \geq y\).

Let \(A=\\{4,5,6\\}\) and \(B=\\{5,6,7\\}\) and define binary relations \(R, S\), and \(T\) from \(A\) to \(B\) as follows: For all \((x, y) \in A \times B, \quad(x, y) \in R \quad \Leftrightarrow \quad x \geq y .\) For all \((x, y) \in A \times B, \quad x S y \Leftrightarrow 2 \mid(x-y)\). \(T=\\{(4,7),(6,5),(6,7)\\} .\) a. Draw arrow diagrams for \(R, S\), and \(T\). b. Indicate whether any of the relations \(R, S\), and \(T\) are functions.

a. Find all binary relations from \(\\{0,1\\}\) to \(\\{1\\}\). b. Find all functions from \(\\{0,1\\}\) to \(\\{1\\}\). c. What fraction of the binary relations from \(\\{0,1\\}\) to \(\\{1\\}\) are functions?

Let \(A=\\{a, b\\}\). Describe all partial order relations on \(A\).

Determine which of the following congruence relations are true and which are false. a. \(17=2(\bmod 5)\) b. \(4=-5(\bmod 7)\) c. \(-2 \equiv-8(\bmod 3)\) d, \(-6 \equiv 22(\bmod 2)\)

a. Let \(R\) be the relation of congruence modulo 3 . Which of the following equivalence classes are equal? \([7],[-4],[-6],[17],[4],[27],[19]\) b. Let \(R\) be the relation of congruence modulo 7. Which of the following equivalence classes are equal? $$ [35],[3],[-7],[12],[0],[-2],[17] $$

Suppose \(A\) is a set with \(m\) elements and \(B\) is a set with \(n\) elements. a. How many binary relations are there from \(A\) to \(B\) ? Explain. b. How many functions are there from \(A\) to \(B\) ? Explain. c. What fraction of the binary relations from \(A\) to \(B\) are functions?