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

\[\text{Let a}, b \in A . \text{Then}, \] 
\[a * b = b \] 
\[b * a = a\] 
\[\text{Therefore},\] 
\[a * b \neq b * a\]

Thus, * is not commutative on A.

Associativity:

\[\text{Let } a, b, c \in A . \text{Then}, \] 
\[a * \left( b * c \right) = a * c\] 
\[ = c\] 
\[\left( a * b \right) * c = b * c\] 
\[ = c\] 
\[\text{Therefore},\] 
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in A\]

Thus, * is associative on A.