A binary operation \Delta defined on the set of rational numbers \mathbb{Q} is given by a … — JAMB UTME

A binary operation $\Delta$ defined on the set of rational numbers $\mathbb{Q}$ is given by $a \Delta b = \frac{ab}{4}$ . What is the inverse element of $6$ under this operation?

A
$\frac{2}{3}$
$\frac{8}{3}$ Answer
C
$\frac{1}{6}$
D
$\frac{3}{8}$

Answer

The inverse element of

6

under the given binary operation is

\frac{8}{3}

To find the inverse of an element under a binary operation, the identity element

e

must first be found using

a \Delta e = a

, which yields

\frac{ae}{4} = a \implies e = 4

. Then, setting

6 \Delta x = 4

gives

\frac{6x}{4} = 4

, which simplifies to

3x = 8

, giving the inverse

x = \frac{8}{3}

Step-by-Step Solution

Find the identity element

e

of the binary operation.

e = 4

By definition of identity element,

a \Delta e = a

. Substituting into the definition gives

\frac{ae}{4} = a \implies ae = 4a \implies e = 4

Set up the inverse equation for the element

6

6 \Delta x = 4

, where

x

is the inverse of

6

By definition of inverse element,

a \Delta a^{-1} = e

Solve for the inverse

x

x = \frac{8}{3}

Applying the operation rule:

\frac{6x}{4} = 4 \implies \frac{3x}{2} = 4 \implies 3x = 8 \implies x = \frac{8}{3}

Key Concept

Identity and Inverse Elements of a Binary Operation

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