Question

The following table defines an operation * on the set A={a,b}

Answer 1

(a) If I'm reading the table correctly, by definition of *, we have

a * a = b

a * b = b

b * a = a

b * b = a

(b) Yes, * is a binary operation. A binary operation is a function * that maps elements from A × A to A, where

A × A = {{a, b} | a ∈ A and b ∈ A}

In this case,

A × A = {{a, a}, {a, b}, {b, a}, {b, b}}

and * is defined such that each of these pairs gets mapped to either a or b, both elements of A. In other words, A is closed under *.

(c) The domain is A × A and the co-domain is A.