Question

On the set of integers a binary operation * is defined by a * b = a + b + 2. Write the inverse of 4.

Answer 1

To find the identity element, let e be the identity element in Z with respect to * such that

$$\begin{gathered} a*e=a=e*a,\forall a\in Z \\ a*e=a\text{and}e*a=a,\forall a\in Z \\ \mathrm{Then}, \\ a+e+2=a\text{and}e+a+2=a,\forall a\in Z \\ \Rightarrow e=-2,\forall a\in Z \end{gathered}$$

Thus, $-$2 is the identity element in Z with respect to *.

Let b be the inverse of 4.

$$\begin{gathered} \text{Then,} \\ 4*b=e=b*4 \\ 4*b=e\text{and}b*4=e \\ \Rightarrow 4+b+2=-2\text{and}b+4+2=-2 \\ b=-8\in Z \\ \text{Thus,}-8\text{is the inverse of}4. \end{gathered}$$