91Ó°ÊÓ

Define a relation \(R\) on \(\mathbb{Z}\) as \(x R y\) if and only if \(x^{2}+y^{2}\) is even. Prove \(R\) is an equivalence relation. Describe its equivalence classes.

Define a relation on \(\mathbb{Z}\) as \(x R y\) if \(|x-y|<1 .\) Is \(R\) reflexive? Symmetric? Transitive? If a property does not hold, say why. What familiar relation is this?

Let \(A=\\{1,2,3,4,5,6\\} .\) How many different relations are there on the set \(A\) ?

Prove or disprove: If \(R\) is an equivalence relation on an infinite set \(A,\) then \(R\) has infinitely many equivalence classes.

Prove or disprove: If a relation is symmetric and transitive, then it is also reflexive.

Modifying Exercise 8 (above) slightly, define a relation \(\sim\) on \(\mathbb{Z}\) as \(x \sim y\) if and only if \(|x-y| \leq 1 .\) Say whether \(\sim\) is reflexive. Is it symmetric? Transitive?