Question

Match the following binary operations on the set of Real numbers with their identities.

• A. 3
• B. 5
• C. $\frac{1}{3}$
• D. -2

Answer 1

Consider the given operations:
a) $a\ast b=(a+b)-3$
If e is the identity, then $a\ast e=a$
$\Rightarrow a+e-3=a$
$\Rightarrow e=3$

b) $a\ast b=\frac{ab}{5}$
If e is the identity, then $a\ast e=a$
$\Rightarrow \frac{ae}{5}=a$
$\Rightarrow e=5$

c) $a\ast b=3ab$
If e is the identity, then $a\ast e=a$
$\Rightarrow 3ae=a$
$\Rightarrow e=\frac{1}{3}$

d) $a\ast b=a+b+2$
If e is the identity, then $a\ast e=a$
$\Rightarrow a+e+2=a$
$\Rightarrow e=-2$