Question

Find the identity element in the set ${I^ + }$ of all positive integers defined by $a*b = a + b$ for all a, b $ \in $ ${I^ + }$.

Answer 1

Hint- Identity element is an element of a set which, if combined with another element by a specified binary operation, leaves that element unchanged.

Complete step-by-step answer:
Let us consider e to be an identity element in ${I^ + }$ with respect to * such that
      a*e=e*a=a,$\forall a \in {I^ + }$

$ \Rightarrow $ a*e=a and e*a=a, $\forall a \in {I^ + }$

$ \Rightarrow $a+e=a=e+a $\forall a \in {I^ + }$

$ \Rightarrow $a+e=a and e+a=a, $\forall a \in {I^ + }$

$ \Rightarrow $e=0,$\forall a \in {I^ + }$

So, from the above equations we can conclude and say that 0 is the identity element in ${I^ + }$ with respect to *

Note: An identity element, or neutral element, is a special type of element of a set with respect to a binary operation on that set, which leaves any element of the set unchanged when combined with it.
Make use of the appropriate properties in accordance to the question which is given and obtain the result which is needed.