91Ó°ÊÓ

Consider the relation \(R=\\{(a, a),(b, b),(c, c),(d, d),(a, b),(b, a)\\}\) on set \(A=\\{a, b, c, d\\}\) Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why.

Let \(A=\\{0,1,2,3,4,5\\} .\) Write out the relation \(R\) that expresses \(>\) on \(A .\) Then illustrate it with a diagram.

Let \(A=\\{1,2,3,4,5,6\\}\). Write out the relation \(R\) that expresses | (divides) on \(A\). Then illustrate it with a diagram.

List all the partitions of the set \(A=\\{a, b, c\\}\). Compare your answer to the answer to Exercise 6 of Section 11.3 .

Describe the partition of \(\mathbb{Z}\) resulting from the equivalence relation \(\equiv(\bmod 4)\)

Suppose \(P\) is a partition of a set \(A .\) Define a relation \(R\) on \(A\) by declaring \(x R y\) if and only if \(x, y \in X\) for some \(X \in P\). Prove \(R\) is an equivalence relation on \(A\). Then prove that \(P\) is the set of equivalence classes of \(R\).

Congruence modulo 5 is a relation on the set \(A=\mathbb{Z} .\) In this relation \(x R y\) means \(x \equiv y(\bmod 5) .\) Write out the set \(R\) in set-builder notation.

Consider the relation \(R=\\{(x, x): x \in \mathbb{Z}\\}\) on \(\mathbb{Z}\). Is this \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this?

Suppose \([a],[b] \in \mathbb{Z}_{6}\) and \([a] \cdot[b]=[0]\). Is it necessarily true that either \([a]=[0]\) or \([b]=[0] ?\) What if \([a],[b] \in \mathbb{Z}_{7} ?\)

There are 16 possible different relations \(R\) on the set \(A=\\{a, b\\} .\) Describe all of them. (A picture for each one will suffice, but don't forget to label the nodes.) Which ones are reflexive? Symmetric? Transitive?