Let S = {1, 2, 3, 4}.Then the number of elements in the set {f : S × S → S : f is onto and f(a,b)=f(b,a) ≥ a ∀ (a, b)∈ S × S is _____.
Correct Answer: 37
Solution and Explanation
The correct answer is 37
There are 16 ordered pairs in S × S. We write all these ordered pairs in 4 sets as follows.
A = {(1, 1)}
B = {(1, 4), (2, 4), (3, 4) (4, 4), (4, 3), (4, 2), (4, 1)}
C = {(1, 3), (2, 3), (3, 3), (3, 2), (3, 1)}
D = {(1, 2), (2, 2), (2, 1)}
All elements of set B have image 4 and only element of A has image 1.
All elements of set C have image 3 or 4 and all elements of set D have image 2 or 3 or 4.
We will solve this question in two cases.
Case I : When no element of set C has image 3.
Number of onto functions = 2 (when elements of set D have images 2 or 3)
Case II : When atleast one element of set C has image 3.
Number of onto functions = (23 – 1)(1 + 2 + 2) = 35
Therefore , total number of functions = 37
Top Questions on sets
- If \( R \) is the smallest equivalence relation on the set \( \{1, 2, 3, 4\} \) such that \( \{(1,2), (1,3)\} \subseteq R \), then the number of elements in \( R \) is ______.
- Number of subsets of set of letters of word 'MONOTONE' is:
- In a statistical investigation of 1003 families of Calcutta, it was found that 63 families have neither a radio nor a TV, 794 families have a radio, and 187 have a TV. The number of families having both a radio and a TV is:
- If \( a>0, b>0, c>0 \) and \( a, b, c \) are distinct, then \( (a + b)(b + c)(c + a) \) is greater than:
- The modulus of the complex number \( z \) such that \( |z + 3 - i| = 1 \) and \( \arg(z) = \pi \) is equal to:
Questions Asked in JEE Main exam
- Two tuning forks \(A\) and \(B\) are sounded together giving rise to 8 beats in 2 s. When fork \(A\) is loaded with wax, the beat frequency is reduced to 4 beats in 2 s. If the original frequency of tuning fork \(B\) is \(380\ \text{Hz}\), find the original frequency of tuning fork \(A\). Given: \[ f_B = 380\ \text{Hz} \]
- JEE Main - 2026
- General Physics
Let the lines $L_1 : \vec r = \hat i + 2\hat j + 3\hat k + \lambda(2\hat i + 3\hat j + 4\hat k)$, $\lambda \in \mathbb{R}$ and $L_2 : \vec r = (4\hat i + \hat j) + \mu(5\hat i + + 2\hat j + \hat k)$, $\mu \in \mathbb{R}$ intersect at the point $R$. Let $P$ and $Q$ be the points lying on lines $L_1$ and $L_2$, respectively, such that $|PR|=\sqrt{29}$ and $|PQ|=\sqrt{\frac{47}{3}}$. If the point $P$ lies in the first octant, then $27(QR)^2$ is equal to}
- JEE Main - 2026
- Three Dimensional Geometry
- Let \[ A=\{z\in\mathbb{C}:|z-2|\le 4\} \quad\text{and}\quad B=\{z\in\mathbb{C}:|z-2|+|z+2|=5\}. \] Then the maximum value of \(|z_1-z_2|\), where \(z_1\in A\) and \(z_2\in B\), is:
- JEE Main - 2026
- Miscellaneous
- Let \(y=y(x)\) be the solution of the differential equation \[ x\frac{dy}{dx}=y-x^2\cot x,\quad x\in(0,\pi) \] If \(y\!\left(\frac{\pi}{2}\right)=\frac{\pi^2}{2}\), then \[ 6y\!\left(\frac{\pi}{6}\right)-8y\!\left(\frac{\pi}{4}\right) \] is equal to:
- JEE Main - 2026
- Miscellaneous
- The sum of the coefficients of \(x^{499}\) and \(x^{500}\) in \[ (1+x)^{1000}+x(1+x)^{999}+x^2(1+x)^{998}+\cdots+x^{1000} \] is:
- JEE Main - 2026
- Binomial theorem
Concepts Used:
Sets
In mathematics, a set is a well-defined collection of objects. Sets are named and demonstrated using capital letter. In the set theory, the elements that a set comprises can be any sort of thing: people, numbers, letters of the alphabet, shapes, variables, etc.
Read More: Set Theory
Elements of a Set:
The items existing in a set are commonly known to be either elements or members of a set. The elements of a set are bounded in curly brackets separated by commas.
Read Also: Set Operation
Cardinal Number of a Set:
The cardinal number, cardinality, or order of a set indicates the total number of elements in the set.
Read More: Types of Sets