A comprehensive platform for Teachers to create standard question papers and Students to practice Case-Based, Assertion-Reason, and Critical Thinking questions.
Create professional PDF/Word papers with logo, instructions, and mixed question types in minutes.
Explore our repository by Class and Topic. Filter by "Knowledge" or "Competency" levels.
For Students. Take timed MCQ tests to check your understanding. Get instant feedback.
According to NEP 2020, rote learning is out. The focus has shifted to assessing a student's ability to apply concepts in real-life situations.
Questions derived from real-world passages to test analytical skills.
Testing the logic behind concepts, not just the definition.
Open-ended scenarios that require thinking beyond the textbook.
We provide complete AI-Powered Explanations for every question.
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.
We can list the pairs (x, y) that satisfy this condition:
Therefore, 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.
By counting the pairs listed above, we find that there are 7 pairs where max{x, y} = 3 and 9 pairs where max{x, y} = 4. Therefore, the total number of elements in R is 7 + 9 = 16.
Thus, statement (S₁) is false, since it claims that the number of elements in R is 18.
Step 3: Check if R is symmetric.
A relation is symmetric if whenever (x, y) ∈ R, then (y, x) ∈ R. From the definition of R, max{x, y} = max{y, x}. Therefore, if (x, y) ∈ R, then max{x, y} ∈ {3, 4}, which implies max{y, x} ∈ {3, 4}, and thus (y, x) ∈ R. So, R is symmetric.
Step 4: Check if R is reflexive.
A relation is reflexive if (x, x) ∈ R for all x ∈ A. In this case, we need to check if max{x, x} ∈ {3, 4} for all x ∈ A.
For x = 0, max{0, 0} = 0, which is not in {3, 4}. Therefore, (0, 0) ∉ R, and R is not reflexive.
Step 5: Check if R is transitive.
A relation is transitive if whenever (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, so both pairs are in R. However, max{0, 4} = 4, so (0, 4) ∈ R.
Consider (0,3) ∈ R and (3,0) ∈ R. We need to check if (0,0) ∈ R. Since max{0,0} = 0 which is not in {3,4}, (0,0) is not in R.
Consider (3,0) ∈ R and (0,4) ∈ R. We need to check if (3,4) ∈ R. Since max{3,4} = 4 which is in {3,4}, (3,4) is in R.
Consider (1,3) ∈ R and (3,2) ∈ R. We need to check if (1,2) ∈ R. Since max{1,2} = 2 which is not in {3,4}, (1,2) is not in R. Thus R is not transitive.
Therefore, R is not transitive.
Step 6: Evaluate the statements.
(S₁) is false because the number of elements in R is 16, not 18.
(S₂) is true because R is symmetric, not reflexive, and not transitive.
Correct Answer: only (S₂) is true<\/strong>