Question 1: Find the identity element in the set $I^+$ of all positive integers defined by $a \ast b = a +b$ for all, $a, b \in l^+.$

Given that binary operation $' \ast '$ is valid for set $I^{+}$ of all positive integers defined by $a \ast b = a + b \text{ for all } a,b \in I^+.$

Let us assume $a \in I^+$ and the identity element that we need to compute be $e \in I^+.$

We know that the Identity property is defined as follows: $\Rightarrow a \ast e = e \ast a = a$ $\Rightarrow a + e = a$ $\Rightarrow e = a - a$ $\Rightarrow e = 0$

Therefore the required Identity element w.r.t $\ast$  is 0 $\\$

Question 2: Find the identity element in the set of all rational numbers except $- 1$ with respect to $\ast$ defined by $a \ast b = a+b+ab.$

Given that binary operation $\ast$  is valid for a set of all rational numbers $Q$ defined by $a \ast b = a + b + ab \text{ for all } a,b \in R.$

Let us assume $a \in R$ and the identity element that we need to compute is $e \in R.$

We know that the Identity property is defined as follows: $\Rightarrow a \ast e = e \ast a = a$ $\Rightarrow a + e + ea = a$ $\Rightarrow e + ae = a - a$ $\Rightarrow e(1 + a) = 0$ $\Rightarrow e = 0$

Therefore the required Identity element w.r.t $\ast$  is 0. $\\$

Question 3: If the binary operation $\ast$ on the set $Z$ is defined by $a \ast b =a \ast b -5,$ then element with respect to $\ast .$   [CBSE 2012]

Given that binary operation $\ast$ is valid for the set $Z$ defined by $a \ast b = a + b - 5 \text{ for all } a,b \in Z.$

Let us assume $a \in Z$ and the identity element that we need to compute is $e \in Z.$

We know that he Identity property is defined as follows: $\Rightarrow a \ast e = e \ast a = a$ $\Rightarrow a + e - 5 = a$ $\Rightarrow e - 5 = a - a$ $\Rightarrow e = 5$

Therefore the required Identity element w.r.t $\ast$  is 5. $\\$

Question 4: On the set $Z$ of integers, if the binary operation $\ast$ is defined by $a \ast b = a +b + 2,$ then find the identity element.     [CBSE 2012]

Given that binary operation $\ast$ is valid for the set $Z$ of integers defined by $a \ast b = a + b \text{ for all } a,b \in Z.$

Let us assume $a \in Z$ and the identity element that we need to compute be $e \in Z.$

We know that he Identity property is defined as follows: $\Rightarrow a \ast e = e \ast a = a$ $\Rightarrow a + e + 2 = a$ $\Rightarrow e + 2 = a - a$ $\Rightarrow e = - 2$

Therefore the required Identity element w.r.t $\ast \text{ is } - 2.$