Question

On the set Z of all integers a binary operation * is defined by a * b = a + b + 2 for all a, b ∈ Z. 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

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

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

Now,

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