Question

Let * be a binary operation on Z defined by
a * b = a + b − 4 for all a, b ∈ Z
(i) Show that '*' is both commutative and associative.
(ii) Find the identity element in Z.
(iii) Find the invertible elements in Z.

Answer 1

(i) Commutativity:
$$\begin{gathered} \text{Let}a,b\in Z.\mathrm{Then}, \\ a*b=a+b-4 \\ =b+a-4 \\ =b*a \\ \text{Therefore,} \\ a*b=b*a,\forall a,b\in Z \end{gathered}$$
Thus, * is commutative on Z.

Associativity:
$$\begin{gathered} \text{Let}a,b,c\in Z.\mathrm{Then}, \\ a*\left(b*c\right)=a*\left(b+c-4\right) \\ =a+b+c-4-4 \\ =a+b+c-8 \\ \left(a*b\right)*c=\left(a+b-4\right)*c \\ =a+b-4+c-4 \\ =a+b+c-8 \\ \text{Therefore,} \\ a*\left(b*c\right)=\left(a*b\right)*c,\forall a,b,c\in Z \end{gathered}$$
Thus, * is associative on Z.

(ii) Let e be the identity element in Z with respect to * such that
$$\begin{gathered} a*e=a=e*a,\forall a\in Z \\ a*e=a\text{and}e*a=a,\forall a\in Z \\ a+e-4=a\text{and}e+a-4=a,\forall a\in Z \\ e=4,\forall a\in Z \end{gathered}$$
Thus, 4 is the identity element in Z with respect to *.

$$\begin{gathered} \left(\mathrm{iii}\right)\text{Let}a\in Z\text{and}b\in Z\text{be the inverse of}a.\text{Then,} \\ a*b=e=b*a \\ a*b=e\text{and}b*a=e \\ a+b-4=4\text{and}b+a-4=4 \\ b=8-a\in Z \\ \text{Thus,}8-a\text{is the inverse of}a\in Z. \end{gathered}$$