Maths Q1 – JEE-MAIN 2025

Let

A = {0, 1, 2, 3, 4, 5}

. Let

R

be a relation on

A

defined by

(x, y) \in R

if and only if

max {x, y} \in {3, 4}

. Then among the statements

(S_{1})

: The number of elements in

R

18

, and

(S_{2})

: The relation

R

is symmetric but neither reflexive nor transitive

Clue 1 of 3: Understand the Relation Definition

The relation $R$ includes pairs $(x, y)$ where the maximum of $x$ and $y$ is either 3 or 4.

🥷

Ninja Strategy•Property Verification

Systematically verify each property (number of elements, reflexivity, symmetry, transitivity) for the given relation to determine the truth value of each statement.

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Step 1: Determine the elements of relation R✦ Active

The set is $A = {0, 1, 2, 3, 4, 5}$ . The relation $R$ is defined by $(x, y) \in R$ if $max {x, y} \in {3, 4}$ . This means $max {x, y} = 3$ or $max {x, y} = 4$ .

Case 1: $max {x, y} = 3$ . This implies $x \leq 3$ and $y \leq 3$ , with at least one of $x, y$ being 3. The elements involved are from ${0, 1, 2, 3}$ . The total pairs $(x, y)$ where $x, y \in {0, 1, 2, 3}$ is $4 \times 4 = 16$ . The pairs where $max {x, y} < 3$ (i.e., $x, y \in {0, 1, 2}$ ) are $3 \times 3 = 9$ . So, the number of pairs with $max {x, y} = 3$ is $16 - 9 = 7$ . These pairs are $(0, 3), (1, 3), (2, 3), (3, 0), (3, 1), (3, 2), (3, 3)$ .

Case 2: $max {x, y} = 4$ . This implies $x \leq 4$ and $y \leq 4$ , with at least one of $x, y$ being 4. The elements involved are from ${0, 1, 2, 3, 4}$ . The total pairs $(x, y)$ where $x, y \in {0, 1, 2, 3, 4}$ is $5 \times 5 = 25$ . The pairs where $max {x, y} < 4$ (i.e., $x, y \in {0, 1, 2, 3}$ ) are $4 \times 4 = 16$ . So, the number of pairs with $max {x, y} = 4$ is $25 - 16 = 9$ . These pairs are $(0, 4), (1, 4), (2, 4), (3, 4), (4, 0), (4, 1), (4, 2), (4, 3), (4, 4)$ .

The total number of elements in $R$ is the sum of elements from Case 1 and Case 2, as they are mutually exclusive: $| R | = 7 + 9 = 16$ . Therefore, statement $(S_{1})$ is false.

Step 2: Check properties of relation R (Reflexivity, Symmetry, Transitivity)○ Expand

Reflexivity: A relation is reflexive if $(a, a) \in R$ for all $a \in A$ . For $(a, a) \in R$ , we need $max {a, a} = a \in {3, 4}$ . This means only $(3, 3)$ and $(4, 4)$ are in $R$ . Since $(0, 0) \notin R$ , $(1, 1) \notin R$ , $(2, 2) \notin R$ , $(5, 5) \notin R$ , the relation $R$ is not reflexive.

Symmetry: A relation is symmetric if whenever $(x, y) \in R$ , then $(y, x) \in R$ . If $(x, y) \in R$ , then $max {x, y} \in {3, 4}$ . Since $max {x, y} = max {y, x}$ , it follows that $max {y, x} \in {3, 4}$ , so $(y, x) \in R$ . Thus, $R$ is symmetric.

Transitivity: A relation is transitive if whenever $(x, y) \in R$ and $(y, z) \in R$ , then $(x, z) \in R$ . Consider $x = 0, y = 3, z = 0$ . We have $(0, 3) \in R$ (since $max {0, 3} = 3$ ) and $(3, 0) \in R$ (since $max {3, 0} = 3$ ). For transitivity, $(0, 0)$ must be in $R$ . However, $max {0, 0} = 0 \notin {3, 4}$ , so $(0, 0) \notin R$ . Therefore, $R$ is not transitive.

Based on these checks, statement $(S_{2})$ claims that $R$ is symmetric but neither reflexive nor transitive. All parts of this statement are true. Therefore, statement $(S_{2})$ is true.

Step 3: Conclusion○ Expand

Statement $(S_{1})$ is false (number of elements is 16, not 18).

Statement $(S_{2})$ is true (relation is symmetric, not reflexive, and not transitive).

Thus, only $(S_{2})$ is true.

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Maths Question 1 – JEE-MAIN 2025

Clue 1 of 3: Understand the Relation Definition

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