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

A binary operation $\Delta$ defined on the set of real numbers $\mathbb{R}$ is given by $a \Delta b = a + b - \frac{ab}{4}$ . What is the inverse of $2$ under this operation?

$-4$ Answer
B
$4$
C
$-2$
D
$-\frac{4}{3}$

Answer

The inverse of

2

under the binary operation is

-4

To find the inverse of

2

, we first determine the identity element

e

using

a \Delta e = a

, which yields

e = 0

. Setting

2 \Delta x = 0

gives

2 + x - \frac{2x}{4} = 0

, which simplifies to

2 + \frac{x}{2} = 0

, resulting in

x = -4

Step-by-Step Solution

Determine the identity element

e

of the operation.

e = 0

By definition,

a \Delta e = a

. Thus,

a + e - \frac{ae}{4} = a \implies e\left(1 - \frac{a}{4}\right) = 0

, which gives

e = 0

for all real

a

Set up the inverse equation for the element

2

2 + x - \frac{2x}{4} = 0

Let

x

be the inverse of

2

. By definition,

2 \Delta x = e

, where

e = 0

Simplify and solve for

x

x = -4

Simplify

2 + x - \frac{x}{2} = 0 \implies 2 + \frac{x}{2} = 0 \implies \frac{x}{2} = -2 \implies x = -4

Key Concept

Inverse Element in Binary Operations

Question

Answer

Step-by-Step Solution

Key Concept

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