Class 12 Mathematics Relation and Function Binary Operation

Medium

Consider a binary operation $$ on the set $\{1, 2, 3, 4, 5\}$ given by the following multiplication table. Compute $(2 \,^ \,3) \,^* \,(4 \,^* \,5)$.
(Hint: use the following table)

$^*$ $1$ $2$ $3$ $4$ $5$

$1$ $1$ $1$ $1$ $1$ $1$

$2$ $1$ $2$ $2$ $2$ $2$

$3$ $1$ $2$ $3$ $3$ $3$

$4$ $1$ $2$ $3$ $4$ $4$

$5$ $1$ $2$ $3$ $4$ $5$

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Solution

(B) From the given table:
$1$. Find the value of $(2 \,^* \,3)$. Looking at the row for $2$ and column for $3$,we get $(2 \,^* \,3) = 2$.
$2$. Find the value of $(4 \,^* \,5)$. Looking at the row for $4$ and column for $5$,we get $(4 \,^* \,5) = 4$.
$3$. Now,compute the final expression: $(2 \,^* \,3) \,^* \,(4 \,^* \,5) = 2 \,^* \,4$.
$4$. Looking at the row for $2$ and column for $4$ in the table,we get $2 \,^* \,4 = 2$.
Thus,the final result is $2$.

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Relation and Function

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Binary Operation

Let $^$ be the binary operation on $N$ given by $a ^ b = \text{L.C.M. of } a \text{ and } b$. Find which elements of $N$ are invertible for the operation $^*$?

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Let $P$ be the set of all subsets of a given set $X$. Show that $\cup: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cup B$ and $\cap: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cap B$ are binary operations on the set $P$.

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Let $A = \{0, 1, 2, 3, 4, 5, 6\}$. If $a, b \in A$,and $a * b$ is defined as the remainder when $ab$ is divided by $7$,then find the inverse of $2$ under the operation $*$.

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In $P(X)$,the power set of a non-empty set $X$,a binary operation $$ is defined by $A B = A \cup B, \forall A, B \in P(X)$. Under $*$,which of the following statements is true?

EasyKCET 2010

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Let $$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a b = a + ab$ for all $a, b \in Q$ is commutative.

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$^*$	$1$	$2$	$3$	$4$	$5$
$1$	$1$	$1$	$1$	$1$	$1$
$2$	$1$	$2$	$2$	$2$	$2$
$3$	$1$	$2$	$3$	$3$	$3$
$4$	$1$	$2$	$3$	$4$	$4$
$5$	$1$	$2$	$3$	$4$	$5$

Similar Questions

Let $^*$ be the binary operation on $N$ given by $a ^* b = \text{L.C.M. of } a \text{ and } b$. Find which elements of $N$ are invertible for the operation $^*$?

Let $P$ be the set of all subsets of a given set $X$. Show that $\cup: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cup B$ and $\cap: P \times P \rightarrow P$ given by $(A, B) \rightarrow A \cap B$ are binary operations on the set $P$.

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In $P(X)$,the power set of a non-empty set $X$,a binary operation $*$ is defined by $A * B = A \cup B, \forall A, B \in P(X)$. Under $*$,which of the following statements is true?

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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 which elements of $N$ are invertible for the operation $^*$?

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Let $$ be a binary operation defined on the set of rational numbers $Q$. Determine whether the binary operation defined by $a b = a + ab$ for all $a, b \in Q$ is commutative.