Class 12 Mathematics Relation and Function Binary Operation

Medium

Let $^$ be the binary operation on $N$ given by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

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Solution

(A) For $a, b, c \in N,$
$(a \,^*\, b) \,^*\, c = (\text{L.C.M. of } a \text{ and } b) \,^*\, c = \text{L.C.M. of } a, b, \text{ and } c$
$a \,^*\, (b \,^*\, c) = a \,^*\, (\text{L.C.M. of } b \text{ and } c) = \text{L.C.M. of } a, b, \text{ and } c$
$\therefore (a \,^*\, b) \,^*\, c = a \,^*\, (b \,^*\, c)$
Thus,the operation $^*$ is associative.

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Relation and Function

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Binary Operation

Show that $-a$ is not the inverse of $a \in N$ for the addition operation $+$ on $N$ and $\frac{1}{a}$ is not the inverse of $a \in N$ for the multiplication operation $\times$ on $N$,for $a \neq 1$.

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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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Show that none of the operations given above has an identity element.

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Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a ^ b = a - b$. Determine whether the operation $^*$ is commutative and associative.

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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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Let $^*$ be the binary operation on $N$ given by $a \,^*\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

Similar Questions

Show that $-a$ is not the inverse of $a \in N$ for the addition operation $+$ on $N$ and $\frac{1}{a}$ is not the inverse of $a \in N$ for the multiplication operation $\times$ on $N$,for $a \neq 1$.

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

Show that none of the operations given above has an identity element.

Let $^*$ be a binary operation on the set $Q$ of rational numbers defined as $a ^* b = a - b$. Determine whether the operation $^*$ is commutative and associative.

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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Let $^$ be the binary operation on $N$ given by $a \,^\, b = \text{L.C.M. of } a \text{ and } b$. Is $^*$ associative?

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

Let $^$ be a binary operation on the set $Q$ of rational numbers defined as $a ^ b = a - b$. Determine whether the operation $^*$ is commutative and associative.

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 $N$?