Question

Let * be a binary operation on R defined by a * b = ab + 1. Then, * is
(a) commutative but not associative
(b) associative but not commutative
(c) neither commutative nor associative
(d) both commutative and associative

Answer 1

(a) commutative but not associative

Commutativity:
$$\begin{gathered} \text{Let}a,b\in R \\ a*b=ab+1 \\ =ba+1 \\ =b*a \\ \text{Therefore,} \\ a*b=b*a,\forall a,b\in R \end{gathered}$$

Therefore, * is commutative on R.

Associativity:
$$\begin{gathered} \text{Let}a,b,c\in R \\ a*\left(b*c\right)=a*\left(bc+1\right) \\ =a\left(bc+1\right)+1 \\ =abc+a+1 \\ \left(a*b\right)*c=\left(ab+1\right)*c \\ =\left(ab+1\right)c+1 \\ =abc+c+1 \\ \text{∴}a*\left(b*c\right)\ne \left(a*b\right)*c \\ \text{For example:}\mathit{\text{a=}}\text{1}\mathit{\text{,}}b\mathit{=}2\mathrm{and}c\mathit{=}3\left[\mathrm{which}\mathrm{belong}\mathrm{to}R\right] \\ \mathrm{Now}, \\ 1*\left(2*3\right)=1*\left(6+1\right) \\ =1*7 \\ =7+1 \\ =8 \\ \left(1*2\right)*3=\left(2+1\right)*3 \\ =3*3 \\ =9+1 \\ =10 \\ \Rightarrow 1*\left(2*3\right)\ne \left(1*2\right)*3 \\ \text{Therefore,}\exists \mathit{\text{a=}}\text{1,}\mathrm{b}\mathit{=}2\mathrm{and}\mathrm{c}\mathit{=}3\mathrm{which}\mathrm{belong}\mathrm{to}R\mathrm{such}\mathrm{that}\mathit{}a\mathit{}\mathit{*}\mathit{}\left(b\mathit{}\mathit{*}\mathit{}c\right)\mathit{\ne }\left(a\mathit{}\mathit{*}\mathit{}b\right)\mathit{}\mathit{*}\mathit{}c \end{gathered}$$

Hence, * is not associative on R.