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