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प्रश्न
Check the commutativity and associativity of the following binary operation '*' on R defined by a * b = a + b − 7 for all a, b ∈ R ?
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उत्तर
Commutativity:
\[\text{ Let } a, b \in R . \text{Then}, \]
\[a * b = a + b - 7\]
\[ = b + a - 7\]
\[ = b * a \]
\[\text{Therefore},\]
\[a * b = b * a, \forall a, b \in R\]
Thus, * is commutative on R.
Associativity:
\[\text{ Let } a, b, c \in R . \text{ Then }, \]
\[a * \left( b * c \right) = a * \left( b + c - 7 \right)\]
\[ = a + b + c - 7 - 7\]
\[ = a + b + c - 14\]
\[\left( a * b \right) * c = \left( a + b - 7 \right) * c\]
\[ = a + b - 7 + c - 7\]
\[ = a + b + c - 14\]
\[\text{Therefore},\]
\[a * \left( b * c \right) = \left( a * b \right) * c, \forall a, b, c \in R\]
Thus, * is associative on R.
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