CBSE 2024, Set 3

Question

**Assertion (A):** The relation $R = \{(x,y) : (x+y) 	ext{ is a prime number and } x, y \in N\}$ is not a reflexive relation.

**Reason (R):** The number '$2n$' is composite for all natural numbers $n$. A). Both Assertion (A) and Reason (R) are true and Reason (R) is the correct explanation of the Assertion (A). B). Both Assertion (A) and Reason (R) are true, but Reason (R) is not the correct explanation of the Assertion (A). C). Assertion (A) is true, but Reason (R) is false. D). Assertion (A) is false, but Reason (R) is true.

Edvaya · Accepted Answer

The correct answer is Assertion (A) is true, but Reason (R) is false.. **Solution:**

**Analyze Assertion (A):**
A relation $R$ on a set is reflexive if $(x,x) \in R$ for every element $x$ in the set.
For this specific relation $R$ on natural numbers $N$, $(x,x) \in R$ implies that $(x+x) = 2x$ must be a prime number.
Let's test a few natural numbers:
If $x = 1$, $2(1) = 2$. $2$ is a prime number, so $(1,1) \in R$.
If $x = 2$, $2(2) = 4$. $4$ is not a prime number, so $(2,2) 
otin R$.
Since $(2,2)$ is not in the relation, the relation is **not reflexive**. Thus, Assertion (A) is **True**.

**Analyze Reason (R):**
Reason (R) claims that $2n$ is a composite number for *all* natural numbers $n$.
Let's test $n=1$. $2(1) = 2$. The number $2$ is a prime number; it is not a composite number. Therefore, the statement fails for $n=1$.
Thus, Reason (R) is **False**.

Since (A) is true and (R) is false, the correct option is "Assertion (A) is true, but Reason (R) is false."

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Mathematics > Sets Relations And Functions > Relations And Functions

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