The relation R in R (set of real numbers) is defined by R = \(a,b): a ≤ b^3\, then R is Option 1: Reflexive relation Option 2: Transitive but not symmetric Option 3: Reflexive and transitive but not symmetric Option 4: Neither reflexive nor symmetric nor transitive

Answer: Neither reflexive nor symmetric nor transitive. Solution: Given: R = \(a,b): a ≤ b^3\ where a, b R To determine the properties of relation R, we check if it is reflexive, symmetric, and transitive. For R to be reflexive, (a,a) must be in R for all values of a. This requires a ≤ a^3 for all a R Take a = 0.5: 0.5 ≤ (0.5)^3 0.5 ≤ 0.125 is false. R is not reflexive. For R to be symmetric, if (a,b) R, then (b,a) R. This requires: if a ≤ b^3, then b ≤ a^3. Take a = 0, b = 1: Check (a,b): 0 ≤ 1^3 0 ≤ 1 is true, so (0,1) R Check (b,a): 1 ≤ 0^3 1 ≤ 0 is false, so (1,0) R R is not symmetric. For R to be transitive, if (a,b) R and (b,c) R, then (a,c) R. This requires: if a ≤ b^3 and b ≤ c^3, then a ≤ c^3. Take a = 100, b = 5, c = 2: Check (a,b): 100 ≤ 5^3 100 ≤ 125 is true, so (100, 5) R Check (b,c): 5 ≤ 2^3 5 ≤ 8 is true, so (5, 2) R Check (a,c): 100 ≤ 2^3 100 ≤ 8 is false, so (100, 2) R R is not transitive. R is neither reflexive, nor symmetric, nor transitive.

Algebra

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The relation R in $\mathbb{R}$ (set of real numbers) is defined by $R = \{(a,b): a \leq b^3\}$ , then R is

Reflexive relation

Transitive but not symmetric

Reflexive and transitive but not symmetric

Neither reflexive nor symmetric nor transitive

✅ Correct Option: 4

Given: $R = \{(a,b): a \leq b^3\}$ where $a, b \in \mathbb{R}$

To determine the properties of relation $R$ , we check if it is reflexive, symmetric, and transitive.

For $R$ to be reflexive, $(a,a)$ must be in $R$ for all values of $a$ .

This requires $a \leq a^3$ for all $a \in \mathbb{R}$

Take $a = 0.5$ :

$0.5 \leq (0.5)^3$

$0.5 \leq 0.125$ is false.

$R$ is not reflexive.

For $R$ to be symmetric, if $(a,b) \in R$ , then $(b,a) \in R$ .

This requires: if $a \leq b^3$ , then $b \leq a^3$ .

Take $a = 0, b = 1$ :

Check $(a,b)$ : $0 \leq 1^3$

$0 \leq 1$ is true, so $(0,1) \in R$

Check $(b,a)$ : $1 \leq 0^3$

$1 \leq 0$ is false, so $(1,0) \notin R$

$R$ is not symmetric.

For $R$ to be transitive, if $(a,b) \in R$ and $(b,c) \in R$ , then $(a,c) \in R$ .

This requires: if $a \leq b^3$ and $b \leq c^3$ , then $a \leq c^3$ .

Take $a = 100, b = 5, c = 2$ :

Check $(a,b)$ : $100 \leq 5^3$

$100 \leq 125$ is true, so $(100, 5) \in R$

Check $(b,c)$ : $5 \leq 2^3$

$5 \leq 8$ is true, so $(5, 2) \in R$

Check $(a,c)$ : $100 \leq 2^3$

$100 \leq 8$ is false, so $(100, 2) \notin R$

$R$ is not transitive.

$R$ is neither reflexive, nor symmetric, nor transitive.

2026: 14 May Shift 2

Given the relation $R=\{(1,2),(2,3)\}$ on a set $A=\{1,2,3\}$ , then which of following statements are TRUE?

A. Minimum number of ordered pairs are added to R so that enlarged relation is reflexive is 3

B. Minimum number of ordered pairs are added to R so that enlarged relation is symmetric is 2

C. Minimum number of ordered pairs are added to R so that enlarged relation is an equivalence relation is 7

D. Minimum number of ordered pair are added to R so that enlarged relation is an equivalence relation is 6.

Choose the correct answer from the options given below:

2026: 22 May Shift 1

Let X be a non empty set and S be the collection of all subsets of X. A relation R in S is defined by $R=\{(A,B): A\subseteq B\}$ , where $X\subseteq Y$ means $X$ is proper subset of $Y$ . Then R is:

2022: 17 Aug Shift 2

Let $f : [2, \infty) \to \mathbf{R}$ be a function defined by $f(x) = x^2 - 4x + 5$ . The range of $f$ is :

2022: 16 July Shift 2

A relation R = {(1,1),(2,2),(3,3),(1,2),(2,1),(1,3),(2,3)} on A = {1,2,3} will be an equivalence relation, if we delete:

Choose the correct answer from the options given below:

2022: 25 May Shift 1

The range of the function $f(x) = \frac{1}{3 - \sin 4x}$ is:

2025: 30 May Shift 1

The relation R in the set $\{1, 2, 3\}$ given by $R = \{(1, 1), (2, 2), (3, 3), (1, 2), (1, 3), (2, 3)\}$ is:

The relation R in R\mathbb{R}R (set of real numbers) is defined by R={(a,b):a≤b3}R = \{(a,b): a \leq b^3\}R={(a,b):a≤b3}, then R is

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The relation R in $\mathbb{R}$ (set of real numbers) is defined by $R = \{(a,b): a \leq b^3\}$ , then R is