Question

Which of the following is true?
(a) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Z
(b) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Q
(c) all binary commutative operations are associative
(d) subtraction is a binary operation on N

Answer 1

(b) * defined by $a*b=\frac{a+b}{2}$ is a binary operation on Q.

Let us check each option one by one.

(a)
$$\begin{gathered} \text{If}a=1\mathrm{and}b=2, \\ a*b=\frac{a+b}{2} \\ =\frac{1+2}{2} \\ =\frac{3}{2}\notin Z \end{gathered}$$

Hence, (a) is false.

(b)
$$\begin{gathered} a*b=\frac{a+b}{2}\in Q,\forall a,b\in Q \\ \mathrm{For}\mathrm{example}:\mathrm{Let}a=\frac{3}{2},b=\frac{5}{6}\in Q \\ a*b=\frac{{\displaystyle \frac{3}{2}}+{\displaystyle \frac{5}{6}}}{2} \\ =\frac{9+5}{12} \\ =\frac{14}{12} \\ =\frac{7}{6}\in Q \end{gathered}$$

Hence, (b) is true.

(c)
Commutativity:
$$\begin{gathered} \text{Let}a,b\in N.\mathrm{Then}, \\ a*b=2^{ab} \\ =2^{ba} \\ =b*a \\ \text{Therefore,} \\ a*b=b*a,\forall a,b\in N \end{gathered}$$

Thus, * is commutative on N.

Associativity:
$$\begin{gathered} \text{Let}a,b,c\in N.\mathrm{Then}, \\ a*\left(b*c\right)=a*\left(2^{bc}\right) \\ =2^{a*2^{bc}} \\ \left(a*b\right)*c=\left(2^{ab}\right)*c \\ =2^{ab*2^c} \\ \text{Therefore,} \\ a*\left(b*c\right)\ne \left(a*b\right)*c \end{gathered}$$

Thus, * is not associative on N.
Therefore, all binary commutative operations are not associative.
Hence, (c) is false.

(d) Subtraction is not a binary operation on N because subtraction of any two natural numbers is not always a natural number.
For example: 2 and 4 are natural numbers.
2$-$4 = $-$2 which is not a natural number.
Hence, (d) is false.