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Let \(\ast\) be a binary operation on the set \(Q\) of rational numbers as follows: $(iii)\;\; a \ast b = a+ab$ Find which of the binary operations are commutative and which are associative.

Note: This is part 3 of a 6 part question, split as 6 separate questions here.
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1 Answer

  • An operation $\ast$ on $A$ is commutative if $a\ast b = b \ast a\; \forall \; a, b \in A$
  • An operation $\ast$ on $A$ is associative if $a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A$
Given a binary operation $\ast$ in Q defined by $a \ast b=a+ab$:
$\textbf {Step 1: Checking if the operation is Commutative}$:
For an operation $\ast$ to be commutative $a\ast b = b \ast a$.
$a \ast b = a + ab$ and $b \ast a = b + ab \rightarrow a\ast b \neq b \ast a \rightarrow$ the operation $\ast$ is not commutative.
$\textbf {Step 2: Checking if the operation is Associative}$:
An operation is associative if $a\ast ( b \ast c) = (a \ast b) \ast c$.
$a \ast (b \ast c) = a \ast (b+bc) = a + a(b+bc) = a + ab + abc$
$(a \ast b) \ast c = (a+ab) \ast c = (a+ab) + (a+ab)c = a+ab+ac+abc$
Since $a\ast ( b \ast c) \neq (a \ast b) \ast c$, the operation $\ast$ is not associative.
answered Mar 13, 2013 by sreemathi.v
edited Mar 19, 2013 by balaji.thirumalai

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