Let A be any set containing more than one element. Let '*' be a b… - Vidyadip

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?

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Answer

Commutativity: Let $\text{a, b}\in\text{A.}$ Then,

$\text{a}\ ^*\ \text{b}=\text{b}$

$\text{b}\ ^*\ \text{a}=\text{a}$

Therefore,

$\text{a}\ ^*\ \text{b}\neq\text{b}\ ^*\ \text{a}$

Thus, * is not commutative on A.

Associativity: Let $\text{a, b, c}\in\text{A.}$ Then,

$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=\text{a}\ ^*\ \text{c}$

$=\text{c}$

$(\text{a}\ ^*\ \text{b})\ ^*\ \text{c}=\text{b}\ ^*\ \text{c}$

$=\text{c}$

Therefore,

$\text{a}\ ^*\ (\text{b}\ ^*\ \text{c})=(\text{a}\ ^*\ \text{b})\ ^*\ \text{c},\ \forall\ \text{a, b, c}\in\text{A}$

Thus, * is associative on A.

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