Question

Which of the following is true ?

विकल्प

• * defined by \[a * b = \frac{a + b}{2}\] is a binary operation on Z .

• * defined by \[a * b = \frac{a + b}{2}\]  is a binary operation on Q .

• all binary commutative operations are associative.

• subtraction is a binary operation on N.

Answer 1

Let us check each option one by one.

* defined by \[a * b = \frac{a + b}{2}\] is a binary operation on Z .

\[\text{If }a = 1 \text{ and } b = 2, \]
\[a * b = \frac{a + b}{2}\]
\[ = \frac{1 + 2}{2}\]
\[ = \frac{3}{2} \not\in Z\]

Hence, it is false.

 

* defined by \[a * b = \frac{a + b}{2}\]  is a binary operation on Q .

\[a * b = \frac{a + b}{2} \in Q, \forall a, b \in Q\]
\[\text { For example: Let a } = \frac{3}{2}, b = \frac{5}{6} \in Q\]
\[a * b = \frac{\frac{3}{2} + \frac{5}{6}}{2}\]
\[ = \frac{9 + 5}{12}\]
\[ = \frac{14}{12}\]
\[ = \frac{7}{6} \in Q\]

Hence, it is true.

 

all binary commutative operations are associative.

Commutativity :-

\[\text{ Let a, b } \in N . \text{ Then }, \]
\[a * b = 2^{ab} \]
\[ = 2^{ba} \]
\[ = b * a\]
\[\text{ Therefore },\]
\[a * b = b * a, \forall a, b \in N\]

Thus, * is commutative on N.

Associativity :-

\[\text{ Let a, b, c } \in N . \text{ Then }, \]
\[a * \left( b * c \right) = a * \left( 2^{bc} \right)\]
\[ = 2^{a^* 2^{bc}} \]
\[\left( a * b \right) * c = \left( 2^{ab} \right) * c\]
\[ = 2^{ab^* 2^c} \]
\[\text{ Therefore },\]
\[a * \left( b * c \right) \neq \left( a * b \right) * c\]

Thus, * is not associative on N.

Therefore, all binary commutative operations are not associative.

Hence, it is false.

 

subtraction is a binary operation on N.

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, it is false.