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 JEE Main Mathematics Questions
- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
- If , then the general value of is:
- The general solution of
- If the constant term in the binomial expansion of $\left(\frac{x^{\frac{3}{2}}}{2}-\frac{4}{x^l}\right)^9$ is $-84$ and the coefficient of $x^{-3 l}$ is $2^\alpha \beta$, where $\beta<0$ is an odd number, then $|\alpha l-\beta|$ is equal to_____
Top JEE Main sets Questions
- 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 ______.
Let \(P(S)\) denote the power set of \(S = \{1, 2, 3, \ldots, 10\}\). Define the relations \(R_1\) and \(R_2\) on \(P(S)\) as \(A R_1 B\) if \[(A \cap B^c) \cup (B \cap A^c) = ,\]and \(A R_2 B\) if\[A \cup B^c = B \cup A^c,\]for all \(A, B \in P(S)\). Then:
- Let S = {4, 6, 9} and T = {9, 10, 11, …,1000}. If A = {a1 + a2 + … +ak :k∈N, a1, a2, a3, …, ak∈S}, then the sum of all the elements in the set T – A is equal to ____________.
- Let A = {1, 2, 3, 4, 5, 6, 7} and B = {3, 6, 7, 9}. Then the number of elements in the set {C ⊆ A :Cߏ∩B ≠ φ} is _____________.
- Let R1 = {(a, b) ∈ N × N : |a – b| ≤ 13} and R2 = {(a, b) ∈ N × N : |a – b| ≠ 13}. Then on N:
Top JEE Main Questions
- Two masses $m$ and $\frac{m}{2}$ are connected at the two ends of a massless rigid rod of length $l$. The rod is suspended by a thin wire of torsional constant $k$ at the centre of mass of the rod-mass system(see figure). Because of torsional constant $k$, the restoring torque is $\tau =k\theta$ for angular displacement $\theta$. If the rod is rota ted by $\theta_0$ and released, the tension in it when it passes through its mean position will be:
What will be the equilibrium constant of the given reaction carried out in a \(5 \,L\) vessel and having equilibrium amounts of \(A_2\) and \(A\) as \(0.5\) mole and \(2 \times 10^{-6}\) mole respectively?
The reaction : \(A_2 \rightleftharpoons 2A\)- The value of is
- The number of terms of an is even. The sum of the odd terms is and of the even terms is . The last term exceeds the first by . Then the number of terms in the series is ______.
A black body is at a temperature of 2880 K. The energy of radiation emitted by this body with wavelength between 499 nm and 500 nm is U1, between 999 nm and 1000 nm is U2 and between 1499 nm and 1500 nm is U3. The Wien's constant, b = 2.88×106 nm-K. Then,
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