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Home  >>  CBSE XII  >>  Math  >>  Relations and Functions

Let \(\ast\) be a binary operation on the set \(Q\) of rational numbers as follows: $(vi)\;\; a \ast b = {ab}^2$ Find which of the binary operations are commutative and which are associative.

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

1 Answer

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 $a \ast b=ab^2$:
$\textbf {Step 1: Checking if the operation is Commutative}$:
For an operation $\ast$ to be commutative $a\ast b = b \ast a$.
$ a \ast b = ab^2$, but $b \ast a = ba^2$, which are not equal unless $a=b=1$. Hence the operation $\ast$ is not 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) = a \ast (bc^2) = a(bc^2)^2 = ab^2c^4$ and $(a \ast b) \ast c = (ab^2) \ast c = ab^2c^2$
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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