Question

Find the identity element for the binary operation on set $Q$ of rational numbers defined as follows:
(i)$a * b = {a^2} + {b^2}$
(ii) $a * b = {(a - b)^2}$
(iii) $a * b = a{b^2}$

Answer 1

Hint: Let$e$be the identity element. In each case compute $a * e$ and check the existence of $e$ such that $a * e = a = e * a$ for every $a \in Q$. If the relation holds, then find the value of $e$.


Complete step by step solution:
We are given three binary operations defined on the set $Q$ of rational numbers.
We need to find the identity element in each case.
Let $e$ denote the identity element which is also a rational number.
Let’s recall the definition of an identity element.
For any element $x$ belong to a set with binary operation $ * $, if $e$ denotes the identity element, then we have $x * e = x = e * x$.
(i) Consider the binary operation $a * b = {a^2} + {b^2}$.
Let’s compute $a * e$.
$a * e = {a^2} + {e^2}$
Now, we know that $ + $is a binary operation on $Q$ and $a$ is an element of $Q$. This implies that ${a^2} \in Q$.
But what we need is $a * e = a = e * a$ for every $a \in Q$.
That is we need ${a^2} + {e^2} = a$ for every $a$.
This is not possible.
Therefore, there does not exist an identity element for the binary operation $a * b = {a^2} + {b^2}$.
(ii) Consider $a * b = {(a - b)^2}$
Therefore, we get $a * e = {(a - e)^2} = {a^2} - 2ae + {e^2}$.
We need $a * e = a = e * a$for every$ a \in Q$.
That is we need ${a^2} - 2ae + {e^2} = a$ for every $a$ which is not possible.
Therefore, there does not exist an identity element for the binary operation $a * b = {(a - b)^2}$.

(iii) Consider $a * b = a{b^2}$
We need $a * e = a = e * a$ for every $a \in Q$.
Therefore, we get $a * e = a{e^2}$ and $e * a = e{a^2}$ for every $a \in Q$.
Now, $a{e^2} = e{a^2} \Leftrightarrow a{e^2} - e{a^2} = 0 \Leftrightarrow ae(e - 1) = 0$
Here $a,e,1,0$ are rational numbers.
Therefore, we have $a{e^2} = e{a^2} \Leftrightarrow e(e - 1) = 0$.
Now, this is possible only if $e = 0$ or $e = 1$.
Hence the identity element for the binary operation $a * b = a{b^2}$ is $e = 0$ or $e = 1$.


Note: For the set of rational numbers, 0 is the identity element with respect to the binary operation of addition and 1 is the identity element with respect to the binary operation of multiplication.