Chapter 2

Let \(X=\\{a, b, c\\}\) and \(Y=\\{u, v\\}\). List all the maps from \(X\) to \(Y\) and list all the maps from \(Y\) to \(X\).

Let \(g: X \rightarrow Y\) and \(f: Y \rightarrow Z\) be functions. Show that (a) if \(f\) and \(g\) are both injective then \(f g\) is injective; (b) if \(f\) and \(g\) are both surjective then \(f g\) is surjective. Give examples to show that if \(f\) is injective and \(g\) is surjective then \(f g\) need neither be injective nor surjective.

When \(X=\\{a, b, c\\}\), list all the maps \(f: X \rightarrow X\) which are constant (so that \(f(a)=f(b)=f(c)\) ). Write down the composition table for these maps. Do these maps form a group?

Prove that the relation on the set \(\mathbf{Z}\) defined by \(x R y\) if \(x+y\) is an even integer is an equivalence relation, and determine the equivalence classes. Is the relation \(x R y\) if \(x+y\) is divisible by 3 an equivalence relation?

Write down the addition table for the congruence classes modulo 4 , and the multiplication table for the non-zero congruence classes modulo \(5 .\)

Show that multiplication of congruence classes modulo \(n\) is well-defined.