Question

Find the identity elements with respect to addition in integers?
(a) 1
(b) -1
(c) 0
(d) none

Answer 1

Hint: We start solving the problem by recalling the definition of identity element with respect of addition. We first use the condition I + Z = Z to find the value of the identity element ‘I’. We then use the condition Z + I = Z to find the value of the identity element ‘I’. In the final we both the obtained values of ‘I’ are the same or not.

Complete step-by-step answer:
Given that we need to find the identity element with respect to addition in integers.
We know that an integer which is subjected to any addition operation to another integer to leave the value of another integer unchanged is known as the identity element with respect to addition.
Let us assume the identity element be ‘I’ and an element to represent integers be ‘Z’.
Since, ‘I’ is an identity element it should satisfy I + Z = Z + I = Z.
Let us start solving for I by using I + Z = Z.
So, we have got I + Z = Z.
We have got I = Z – Z.
We have got I = 0 ---(1).
Now, we start solving for I by using Z + I = Z.
So, we have got Z + I = Z.
We have got I = Z – Z.
We have got I = 0 ---(2).
From equations (1) and (2), we have got the value of the identity element when addition with integer is done as 0.
∴ The identity element with respect to addition in integers is 0.
So, the correct answer is “Option C”.

Note: Here we used I + Z = Z + I = Z to find the value of the identity element with respect to addition. Similar way is followed to find the identity element with respect to subtraction and multiplication. ‘0’ is an identity element even with respect to subtraction as we can take +Z as –(–Z).