(B) On the set of positive integers $Z^{+}$,the operation $*$ is defined by $a * b = a - b$. $A$ binary operation $*$ on a set $S$ is a function $*: S

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No,it is not a binary operation because $a - b$ is not always in $Z^{+}$.

(B) On the set of positive integers $Z^{+}$,the operation $*$ is defined by $a * b = a - b$. $A$ binary operation $*$ on a set $S$ is a function $*: S \times S \to S$. For the operation to be binary,for every pair $(a, b) \in Z^{+} \times Z^{+}$,the result $a * b$ must belong to $Z^{+}$. Consider the elements $a = 1$ and $b = 2$,where $1, 2 \in Z^{+}$. Then $a * b = 1 * 2 = 1 - 2 = -1$. Since $-1 \notin Z^{+}$,the operation $*$ is not a binary operation on $Z^{+}$.

Class 12 Mathematics Relation and Function Binary Operation

Medium

Determine whether or not the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a - b$.

A
Yes,it is a binary operation.
B
No,it is not a binary operation because $a - b$ is not always in $Z^{+}$.
C
No,it is not a binary operation because $a - b$ is not commutative.
D
No,it is not a binary operation because $a - b$ is not associative.

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Relation and Function

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Binary Operation

Given a non-empty set $X$,consider the binary operation $^: P(X) \times P(X) \rightarrow P(X)$ defined by $A \,^\, B = A \cap B$ for all $A, B \in P(X)$,where $P(X)$ is the power set of $X$. Show that $X$ is the identity element for this operation and $X$ is the only invertible element in $P(X)$ with respect to the operation.

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Let $^$ be the binary operation on $N$ defined by $a \,^\, b = \text{H.C.F. of } a \text{ and } b$. Is $^$ commutative? Is $^$ associative? Does there exist an identity for this binary operation on $N$?

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Show that $: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a b = a + 2b$ is not commutative.

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Let $^$ be the binary operation on $N$ given by $a \,^ \,b = \text{L.C.M. of } a \text{ and } b$. Find $5 \,^* \,7$ and $20 \,^* \,16$.

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Is $^$ defined on the set $\{1, 2, 3, 4, 5\}$ by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$ a binary operation? Justify your answer.

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Determine whether or not the definition of $*$ given below gives a binary operation. In the event that $*$ is not a binary operation,give justification for this. On $Z^{+}$,define $*$ by $a * b = a - b$.

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Let $^*$ be the binary operation on $N$ defined by $a \,^*\, b = \text{H.C.F. of } a \text{ and } b$. Is $^*$ commutative? Is $^*$ associative? Does there exist an identity for this binary operation on $N$?

Show that $*: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a * b = a + 2b$ is not commutative.

Let $^*$ be the binary operation on $N$ given by $a \,^* \,b = \text{L.C.M. of } a \text{ and } b$. Find $5 \,^* \,7$ and $20 \,^* \,16$.

Is $^*$ defined on the set $\{1, 2, 3, 4, 5\}$ by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$ a binary operation? Justify your answer.

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Determine whether or not the definition of $$ given below gives a binary operation. In the event that $$ is not a binary operation,give justification for this. On $Z^{+}$,define $$ by $a b = a - b$.

Let $^$ be the binary operation on $N$ defined by $a \,^\, b = \text{H.C.F. of } a \text{ and } b$. Is $^$ commutative? Is $^$ associative? Does there exist an identity for this binary operation on $N$?

Show that $: \mathbb{R} \times \mathbb{R} \rightarrow \mathbb{R}$ defined by $a b = a + 2b$ is not commutative.

Let $^$ be the binary operation on $N$ given by $a \,^ \,b = \text{L.C.M. of } a \text{ and } b$. Find $5 \,^* \,7$ and $20 \,^* \,16$.

Is $^$ defined on the set $\{1, 2, 3, 4, 5\}$ by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$ a binary operation? Justify your answer.