Algebra

Relations & Functions

Medium

core

Let N be set of natural numbers, then the function $f: \mathbb{N} \rightarrow \mathbb{N}$ , defined by

$f(x) = \begin{cases} \frac{n+1}{2} & \text{, if } n \text{ is odd} \\ \frac{n}{2} & \text{, if } n \text{ is even} \end{cases}$ , is

injective but not surjective

surjective but not injective

bijective

neither injective nor surjective

✅ Correct Option: 2

The function takes a natural number and:

If $n$ is odd: $f(n) = \frac{n+1}{2}$
If $n$ is even: $f(n) = \frac{n}{2}$

Examples:

$f(1) = \frac{1+1}{2} = 1$

$f(2) = \frac{2}{2} = 1$

$f(3) = \frac{3+1}{2} = 2$

$f(4) = \frac{4}{2} = 2$

$f(5) = \frac{5+1}{2} = 3$

To check if the function is injective, consider whether different inputs can produce the same output.

$f(1) = 1$

$f(2) = 1$

Since $f(1) = f(2)$ but $1 \neq 2$ , the function is not injective.

To check if the function is surjective, determine whether every natural number $k$ can be reached as an output.

For any natural number $k$ , let $n = 2k$ .

Then $f(2k) = \frac{2k}{2} = k$

Examples:

For $k = 1$ : $f(2) = 1$

For $k = 5$ : $f(10) = 5$

For $k = 100$ : $f(200) = 100$

Every natural number is reachable as an output, so the function is surjective.

The function is surjective but not injective.

2025: 22 May Shift 2

The graph given below represents which of the following function?

2025: 16 May Shift 1

Which of the following are correct?

(A) If a ≡ b(mod n), then -a ≡ -b (mod n)

(B) If a + b = c, then a(mod n) + b(mod n) ≡ (a + b + c)(mod n).

(D) If a ≡ b (mod n), then (a + k) ≡ (b + k)(mod n), ∀ k ∈ I.

Choose the correct answer from the options given below:

2025: 26 May Shift 2

Let A = {1, 2, 3}. The number of equivalence relations containing (1, 3) 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:

2026: 18 May Shift 1

If $A = \{x: x \in R, -\frac{\pi}{2} \leq x \leq \frac{\pi}{2}\}$ and $B = \{y: y \in R, -1 \leq y \leq 1\}$ and function $f: A \rightarrow B$ defined by $f(x) = \sin x$ , then $f$ is

2023: 22 May Shift 3

Let R be a relation on the set of natural numbers N defined by nRm if n divides m. Then R is :

(A) Reflexive Relation

(B) Symmetric Relation

(D) Identity Relation

Choose the correct answer from the options given below :

Solution

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