Class 12 Mathematics Relation and Function Binary Operation

Medium

State whether the following statement is true or false. Justify: For an arbitrary binary operation $^$ on a set $N$,$a \,^ \,a = a$ for all $a \in N$.

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Solution

(B) The statement is false.
To justify this,consider a counterexample.
Let the binary operation $^*$ on the set of natural numbers $N$ be defined as $a \,^* \,b = a + b$ for all $a, b \in N$.
Now,consider an element $a = 3 \in N$.
According to the given condition,we should have $3 \,^* \,3 = 3$.
However,using our defined operation,we get $3 \,^* \,3 = 3 + 3 = 6$.
Since $6 \neq 3$,the condition $a \,^* \,a = a$ does not hold for all binary operations on $N$.
Thus,the statement is false.

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Binary Operation

Let $$ be a binary operation defined on $Q$. Determine whether the binary operation defined by $a b = a - b$ for all $a, b \in Q$ is 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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In the group $G=\{0, 1, 2, 3, 4, 5\}$ under addition modulo $6$,$(2 +_{6} 3^{-1} +_{6} 4)^{-1}$ is equal to

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Show that $0$ is the identity for addition on $R$ and $1$ is the identity for multiplication on $R$. But there is no identity element for the operations $-: R \times R \rightarrow R$ and $\div : R_* \times R_* \rightarrow R_*$.

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State whether the following statement is true or false. Justify: For an arbitrary binary operation $^*$ on a set $N$,$a \,^* \,a = a$ for all $a \in N$.

Similar Questions

Let $*$ be a binary operation defined on $Q$. Determine whether the binary operation defined by $a * b = a - b$ for all $a, b \in Q$ is commutative.

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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State whether the following statement is true or false. Justify: For an arbitrary binary operation $^$ on a set $N$,$a \,^ \,a = a$ for all $a \in N$.

Let $$ be a binary operation defined on $Q$. Determine whether the binary operation defined by $a b = a - b$ for all $a, b \in Q$ is 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 $20 \,^* \,16$.