CBSE Class 12 Mathematics Relations and Functions Q #2128
COMPETENCY BASED
APPLY
3 Marks Generic-Curriculum SA
Show that the relation $R$ in the set $R$ of real numbers, defined as $R = \{(a, b) : a \le b^2\}$ is neither reflexive nor symmetric nor transitive. [All India 2024]
Prev Next

AI Tutor Explanation

Powered by Gemini

Detailed Solution

Step 1: Check for Reflexivity

A relation $R$ is reflexive if $(a, a) \in R$ for all $a \in \mathbb{R}$. Here, the condition is $a \le a^2$. Consider $a = 0.5$. Then $0.5 \le (0.5)^2$ becomes $0.5 \le 0.25$, which is false. Thus, $R$ is not reflexive.

Step 2: Check for Symmetry

A relation $R$ is symmetric if $(a, b) \in R \implies (b, a) \in R$. Consider $(1, 2)$. Since $1 \le 2^2$ ($1 \le 4$) is true, $(1, 2) \in R$. However, for $(2, 1)$, we check $2 \le 1^2$, which is $2 \le 1$. This is false. Thus, $R$ is not symmetric.

Step 3: Check for Transitivity

A relation $R$ is transitive if $(a, b) \in R$ and $(b, c) \in R \implies (a, c) \in R$. Consider $(3, 2)$ and $(2, 1.5)$. $$3 \le 2^2 \implies 3 \le 4 \text{ (True)}$$ $$2 \le (1.5)^2 \implies 2 \le 2.25 \text{ (True)}$$ Now check $(3, 1.5)$: $3 \le (1.5)^2 \implies 3 \le 2.25$, which is false. Thus, $R$ is not transitive.

Final Answer: The relation is neither reflexive, nor symmetric, nor transitive.

AI generated content. Review strictly for academic accuracy.

Pedagogical Audit
Bloom's Analysis: This is an APPLY question because the student must apply the definitions of equivalence relations to a specific mathematical set and inequality.
Knowledge Dimension: PROCEDURAL
Justification: The student must follow a step-by-step logical procedure to test three distinct properties using counter-examples.
Syllabus Audit: In the context of CBSE Class 12, this is classified as COMPETENCY. This question tests the conceptual understanding of relations beyond rote memorization by requiring the construction of counter-examples.

More from this Chapter

SA
Let n be a fixed positive integer. A relation R is defined in set Z such that $R=\{(x,y):(x-y) \text{ is divisible by } n, x, y\in Z\}$. Determine if R is an equivalence relation.
VSA
Let $g(x) = x^2 - 4x - 5$, then check whether $f$ is one-one or onto. [All India 2016]
SA
Prove that the function $f$ is surjective, where $f : N \to N$ such that $f(n) = \begin{cases} \frac{n + 1}{2}, & \text{if } n \text{ is odd} \\ \frac{n}{2}, & \text{if } n \text{ is even} \end{cases}$.
SA
If $A = \{x \mid x \in N, x \le 5\}$, $B = \{x \mid x \in Z, x^2 - 5x + 6 = 0\}$, then find the number of onto functions from $A$ to $B$.
LA
Let $f, g : N - \{1\} \to N$ be functions defined by $f(a) = \alpha$, where $\alpha$ is the maximum of the powers of those primes $p$ such that $p^\alpha$ divides $a$, and $g(a) = a + 1$, for all $a \in N - \{1\}$. Then, check whether the function $f + g$ is one-one or onto. [Competency]
View All Questions