Class JEE Mathematics Sets, Relations, and Functions Q #994
COMPETENCY BASED
APPLY
4 Marks 2025 JEE Main 2025 (Online) 8th April Evening Shift MCQ SINGLE
Let A = {0, 1, 2, 3, 4, 5}. Let R be a relation on A defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. Then among the statements

(S1): The number of elements in R is 18, and

(S2): The relation R is symmetric but neither reflexive nor transitive
(A) both are false
(B) only (S1) is true
(C) only (S2) is true
(D) both are true
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Correct Answer: C
Explanation
To evaluate the relation $R$ on the set $A = {0, 1, 2, 3, 4, 5}$, we first need to understand the conditions for an element $(x, y)$ to be in $R$. Specifically, $(x, y) ∈ R$ if and only if $max{x,y} ∈ {3,4}$.

Considering this, let's list the pairs:

For $max{x, y} = 3$, the possible pairs are:

$(0, 3), (3, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3)$

For $max{x, y} = 4$, the possible pairs are:

$(0, 4), (4, 0), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)$

Combining these, the set $R$ consists of the following elements:

$R = {(0, 3), (3, 0), (1, 3), (3, 1), (2, 3), (3, 2), (3, 3), (0, 4), (4, 0), (1, 4), (4, 1), (2, 4), (4, 2), (3, 4), (4, 3), (4, 4)}$

This gives us a total of 16 elements in $R$, not 18 as initially claimed in statement $S1$.

Next, we analyze the properties of the relation $R$:

Reflexivity: A relation is reflexive if $(x, x) ∈ R$ for all $x ∈ A$. For example, $(0, 0), (1, 1), (2, 2)$ are not in $R$, so $R$ is not reflexive.

Symmetry: A relation is symmetric if whenever $(a, b) ∈ R$, then $(b, a) ∈ R$ as well. For all pairs $(x, y)$ listed, both $(x, y)$ and $(y, x)$ are present. Thus, $R$ is symmetric.

Transitivity: A relation is transitive if whenever $(a, b) ∈ R$ and $(b, c) ∈ R$, then $(a, c) ∈ R$. An example where transitivity fails is $(0, 3)$ and $(3, 1)$ are in $R$ but $(0, 1)$ is not in $R$. Therefore, $R$ is not transitive.

In conclusion, statement $S2$ is correct as $R$ is symmetric but neither reflexive nor transitive.

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Step-by-Step Solution

Step 1: Determine the elements of the relation R.

The relation R is defined by (x, y) ∈ R if and only if max{x, y} ∈ {3, 4}. This means that the maximum of x and y must be either 3 or 4.

If max{x, y} = 3, then the possible pairs (x, y) are (0, 3), (1, 3), (2, 3), (3, 3), (3, 0), (3, 1), (3, 2).

If max{x, y} = 4, then the possible pairs (x, y) are (0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (4, 0), (4, 1), (4, 2), (4, 3).

Therefore, the relation R consists of the following pairs:

R = {(0, 3), (1, 3), (2, 3), (3, 3), (3, 0), (3, 1), (3, 2), (0, 4), (1, 4), (2, 4), (3, 4), (4, 4), (4, 0), (4, 1), (4, 2), (4, 3)}

Step 2: Count the number of elements in R.

The number of elements in R is 7 + 9 = 16.

Therefore, statement (S1) is false since it claims that the number of elements in R is 18.

Step 3: Check if R is symmetric.

For R to be symmetric, if (x, y) ∈ R, then (y, x) ∈ R.

If max{x, y} = 3, then max{y, x} = 3. If max{x, y} = 4, then max{y, x} = 4.

Since max{x, y} = max{y, x}, if (x, y) ∈ R, then (y, x) ∈ R. Therefore, R is symmetric.

Step 4: Check if R is reflexive.

For R to be reflexive, (x, x) ∈ R for all x ∈ A.

Consider x = 0. max{0, 0} = 0, which is not in {3, 4}. Therefore, (0, 0) ∉ R.

Thus, R is not reflexive.

Step 5: Check if R is transitive.

For R to be transitive, if (x, y) ∈ R and (y, z) ∈ R, then (x, z) ∈ R.

Consider (0, 3) ∈ R and (3, 4) ∈ R. Then max{0, 3} = 3 and max{3, 4} = 4.

If R is transitive, then (0, 4) should be in R. max{0, 4} = 4, so (0, 4) ∈ R.

However, consider (0, 3) ∈ R and (3, 0) ∈ R. If R is transitive, then (0, 0) ∈ R. But max{0, 0} = 0, which is not in {3, 4}. Therefore, (0, 0) ∉ R.

Consider (3,0) ∈ R and (0,3) ∈ R. If R is transitive, then (3,3) ∈ R. max{3,3} = 3, so (3,3) ∈ R.

Consider (1,3) ∈ R and (3,2) ∈ R. If R is transitive, then (1,2) should be in R. max{1,2} = 2, which is not in {3,4}. Therefore, (1,2) ∉ R.

Thus, R is not transitive.

Step 6: Evaluate the statements.

(S1): The number of elements in R is 18. (False, the number of elements is 16)

(S2): The relation R is symmetric but neither reflexive nor transitive. (True)

Correct Answer: only (S2) is true<\/strong>

AI Suggestion: Option C

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because it requires the student to apply the definition of relations, symmetry, reflexivity, and transitivity to a specific set and relation defined by a maximum function.
Knowledge Dimension: CONCEPTUAL
Justification: The question tests the understanding of the concepts of relations, symmetry, reflexivity, and transitivity, rather than just recalling facts or performing routine procedures.
Syllabus Audit: In the context of JEE, this is classified as COMPETENCY. It requires application of the concepts of relations and functions to determine the properties of a given relation, which goes beyond direct recall from the textbook.

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