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# Let $\ast$ be a binary operation on the set $Q$ of rational numbers as follows: $(v)\;\; a \ast b = \frac{ab}{4}$ Find which of the binary operations are commutative and which are associative.

Note: This is part 5 of a 6 part question, split as 6 separate questions here.

Toolbox:
• 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 $a \ast b= \large \frac{ab}{4}$
$\textbf {Step 1: Checking if the operation is Commutative}$:
For an operation $\ast$ to be commutative $a\ast b = b \ast a$.
$a \ast b = \large \frac{ab}{4}$$= \; b \ast a =$$ \large \frac{ba}{4}$$\rightarrow \ast is commutative. \textbf {Step 2: Checking if the operation is Associative}: For an operation \ast on A is associative a\ast ( b \ast c) = (a \ast b) \ast c\; \forall \; a, b, c \in A (a \ast b ) \ast c = \large \frac{ab}{4}$$ \ast c = \large \frac {abc}{16}$
$a \ast( b \ast c ) = a \ast \large \frac{bc}{4}$$= \large \frac {abc}{16}$
Since $a\ast ( b \ast c) = (a \ast b) \ast c$ the operation $\ast$ is associative
edited Mar 19, 2013