Question

If ‘ $*$ ’ is a binary operation on the set $\mathbb{R}$ of real numbers defined by $a*b=a+b-2$ , find the identity element for the binary operation ′ $*$ ′.

Answer 1

Hint:First, let’s define what a binary operation is, and then after that, we will write the definition of the identity element. Then we will apply the same thing for the given function in the question that is $a*b=a+b-2$ and then finally get the answer.

Complete step by step answer:
Let’s first understand what a binary operation is.
So, basically a binary operation $f\left( x,y \right)$ is an operation that applies to two quantities or expressions $x$ and $y$ . We take the set of numbers on which the binary operations are performed as $A$. The operations (addition, subtraction, division, multiplication, etc.) can be generalized as a binary operation is performed on two elements from setting $A$. The result of the operation on $x$ and $y$ is another element from the same set X.
Thus, A binary operation on non-empty set $A$ is a map $f: A\times A\to A$ such that:
A. $f$ is defined for every elements in $A$and
B. $f$ uniquely associates each pair of elements in $A$ to some element of $A$
Now, let’s see what an identity element is of binary operations.
So, for a binary operation say $*:A\times A\to A$ , $e$ is called identity of $*$ if :
$a*e=e*a=a$ .
Here, $e$ is called the identity element of the binary operation.
Now, we are given ‘ $*$ ’ is a binary operation defined as $a*b=a+b-2$ .
Now, let $e$ be the identity element of binary operation ‘$*$’.
Then , $a*e=a+e-2$, now as we saw above: $a*e=a$ , therefore: $a=a+e-2\Rightarrow e=2$ .
Hence, the identity element is $2$.

Note:
 Typical examples of binary operations are the addition $\left( + \right)$ and multiplication $\left( \times \right)$ of numbers and matrices as well as composition of functions on a single set. For instance, On the set of real numbers R, \[f\left( a,b \right)=a+b\] is a binary operation since the sum of two real numbers is a real number.