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Question
Let * be a binary operation on R defined by a * b = ab + 1. Then, * is _________________ .
Options
commutative but not associative
associative but not commutative
neither commutative nor associative
both commutative and associative
Solution
commutative but not associative
Commutativity:
\[\text { Let } a, b \in R\]
\[a * b = ab + 1\]
\[ = ba + 1\]
\[ = b * a\]
\[\text { Therefore },\]
\[a * b = b * a, \forall a, b \in R\]
Therefore, * is commutative on R.
Associativity:
\[\text{ Let }a, b, c \in R\]
\[a * \left( b * c \right) = a * \left( bc + 1 \right)\]
\[ = a\left( bc + 1 \right) + 1\]
\[ = abc + a + 1\]
\[\left( a * b \right) * c = \left( ab + 1 \right) * c\]
\[ = \left( ab + 1 \right)c + 1\]
\[ = abc + c + 1\]
\[\therefore a * \left( b * c \right) \neq \left( a * b \right) * c\]
\[\text{ For example }:a=1,b = 2 \text{ and } c = 3 \left[ \text{ which belong to R } \right]\]
\[\text{ Now }, \]
\[1 * \left( 2 * 3 \right) = 1 * \left( 6 + 1 \right)\]
\[ = 1 * 7\]
\[ = 7 + 1\]
\[ = 8\]
\[\left( 1 * 2 \right) * 3 = \left( 2 + 1 \right) * 3\]
\[ = 3 * 3\]
\[ = 9 + 1\]
\[ = 10\]
\[ \Rightarrow 1 * \left( 2 * 3 \right) \neq \left( 1 * 2 \right) * 3\]
\[\text { Therefore }, \exists a=1,b = 2 \text{ and } c = 3 \text{ which belong to R such that a } * \left( b * c \right) \neq \left( a * b \right) * c\]
Hence, * is not associative on R.
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