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^+.

Answer:

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

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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.

Answer:

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.

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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]

Answer:

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.

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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]

Answer:

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.