Question

Let A be any set containing more than one element. Let '*' be a binary operation on A defined by

a * b = b for all a, b ∈ A

Is '*' commutative or associative on A?

Answer 1

Commutativity:
$$\begin{gathered} \text{Let}a,b\in A.\mathrm{Then}, \\ a*b=b \\ b*a=a \\ \text{Therefore,} \\ a*b\ne b*a \end{gathered}$$

Thus, * is not commutative on A.

Associativity:
$$\begin{gathered} \text{Let}a,b,c\in A.\mathrm{Then}, \\ a*\left(b*c\right)=a*c \\ =c \\ \left(a*b\right)*c=b*c \\ =c \\ \text{Therefore,} \\ a*\left(b*c\right)=\left(a*b\right)*c,\forall a,b,c\in A \end{gathered}$$

Thus, * is associative on A.