Recent questions tagged sets - GATE CSE Doubts

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TIFR-GS2021 Question
Fix $n\geq6$. Consider the set $C$ of binary strings $x_1x_2...x_n$ of length n such that the bits satisfy the following set of equalities, all modulo 2: $x_i + x_{i+1} + x_i+2 = 0$ for all $1\leq i\leq n-2, x_{n-1} + x_n + x_1 = 0$, and $x_n + x_1 + x_2 = 0$. What ... $n \geq6$ is divisible by $3$ then $|C| = 4$. If $n\geq 6$ is not divisible by $3$ the $|C| =14$

asked Mar 24 in Set Theory & Algebra by zxy123 (3.6k points) | 6 views

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Made easy test series Discrete Mathematics-2
The number of ways of splitting a set of n elements into two parts is Answer is $2^{n-1} – 1$, explanation states this is because partitions can’t be empty, is this true?

asked Jan 29 in Set Theory & Algebra by zxy123 (3.6k points) | 39 views

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Zeal testseries
Are the ordered pairs ($\phi$,$\phi$) , (1,$\phi$) possible?

asked Nov 25, 2020 in Set Theory & Algebra by Kindaichi (10 points) | 15 views

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Zeal test series
Wat will be the value of p :- 1 or 0?

asked Nov 25, 2020 in Set Theory & Algebra by Kindaichi (10 points) | 23 views

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ISI CSB 2018
State, with justification, which of the following expressions f, g and h, define valid real-valued functions over the set of positive rational numbers. We denote a rational number by m/n, where m and n are positive integers. (a) f(m/n) = 2^m − 2^n. (b) g(m/n) = log m − log n. (c) h(m/n) = (m^2 − n^2)/(mn).

asked Aug 20, 2020 in Set Theory & Algebra by suparna kar (9 points) | 32 views

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ISI2015-MCQ-7
Let X be the set {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}. Define the set R by R = {(x, y) ∈ X ×X : x and y have the same remainder when divided by 3}. Then the number of elements in R is (A) 40 (B) 36 (C) 34 (D) 33

asked Jul 2, 2020 in Set Theory & Algebra by suparna kar (9 points) | 38 views

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Show that given relation is an equivalence relation?

asked May 26, 2020 in Set Theory & Algebra by iarnav (231 points) | 63 views

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#DiscreteMaths Relationship between Equivalence classes of an equivalence relationship and partition of a set?

asked May 23, 2020 in Set Theory & Algebra by iarnav (231 points) | 31 views

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GATE2015-1-1‏6 Video Solution
For a set $A$, the power set of $A$ is denoted by $2^{A}$. If $A = \left\{5,\left\{6\right\}, \left\{7\right\}\right\}$, which of the following options are TRUE? $\phi \in 2^{A}$ $\phi \subseteq 2^{A}$ $\left\{5,\left\{6\right\}\right\} \in 2^{A}$ $\left\{5,\left\{6\right\}\right\} \subseteq 2^{A}$ I and III only II and III only I, II and III only I, II and IV only

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 6 views

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GATE2016-2-28 Video Solution
Consider a set $U$ of $23$ different compounds in a chemistry lab. There is a subset $S$ of $U$ of $9$ compounds, each of which reacts with exactly $3$ compounds of $U$. Consider the following statements: Each compound in U \ S reacts with an ... \ S reacts with an even number of compounds. Which one of the above statements is ALWAYS TRUE? Only I Only II Only III None.

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 33 views

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GATE2014-2-50 Video Solution
Consider the following relation on subsets of the set $S$ of integers between 1 and 2014. For two distinct subsets $U$ and $V$ of $S$ we say $U\:<\:V$ if the minimum element in the symmetric difference of the two sets is in $U$. Consider the following two ... $S1$ is true and $S2$ is false $S2$ is true and $S1$ is false Neither $S1$ nor $S2$ is true

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 13 views

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GATE2000-2.6 Video Solution
Let $P(S)$ denotes the power set of set $S.$ Which of the following is always true? $P(P(S)) = P(S)$ $P(S) ∩ P(P(S)) = \{ Ø \}$ $P(S) ∩ S = P(S)$ $S ∉ P(S)$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 7 views

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GATE2015-3-23 Video Solution
Suppose $U$ is the power set of the set $S = \{1, 2, 3, 4, 5, 6\}$. For any $T \in U$, let $|T|$ denote the number of elements in $T$ and $T'$ denote the complement of $T$. For any $T, R \in U \text{ let } T \backslash R$ be the set of all elements in ... $X \backslash Y = \phi)$ $\forall X \in U, \forall Y \in U, (X \backslash Y = Y' \backslash X')$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 8 views

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GATE2017-1-47 Video Solution
The number of integers between $1$ and $500$ (both inclusive) that are divisible by $3$ or $5$ or $7$ is ____________ .

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 7 views

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GATE2005-IT-33 Video Solution
Let $A$ be a set with $n$ elements. Let $C$ be a collection of distinct subsets of $A$ such that for any two subsets $S_1$ and $S_2$ in $C$, either $S_1 \subset S_2$ or $S_2\subset S_1$. What is the maximum cardinality of C? $n$ $n+1$ $2^{n-1} + 1$ $n!$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 7 views

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GATE2006-24 Video Solution
Given a set of elements N = {1, 2, ..., n} and two arbitrary subsets A⊆N and B⊆N, how many of the n! permutations $\pi$ from N to N satisfy min($\pi$(A)) = min($\pi$(B)), where min(S) is the smallest integer in the set of integers S, and $\pi$(S) is the set of integers obtained by ... $n! \frac{|A ∩ B|}{|A ∪ B|}$ $\dfrac{|A ∩ B|^2}{^n \mathrm{C}_{|A ∪ B|}}$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 9 views

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GATE1996-2.4 Video Solution
Which one of the following is false? The set of all bijective functions on a finite set forms a group under function composition. The set $\{1, 2, \dots p-1\}$ forms a group under multiplication mod $p$, where $p$ is a prime number. The set of all strings over a finite ... $\langle G, * \rangle$ if and only if for any pair of elements $a, b \in S, a * b^{-1} \in S$.

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 12 views

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GATE2014-3-16 Video Solution
Let $\Sigma$ be a finite non-empty alphabet and let $2^{\Sigma^*}$ be the power set of $\Sigma^*$. Which one of the following is TRUE? Both $2^{\Sigma^*}$ and $\Sigma^*$ are countable $2^{\Sigma^*}$ is countable and $\Sigma^*$ is uncountable $2^{\Sigma^*}$ is uncountable and $\Sigma^*$ is countable Both $2^{\Sigma^*}$ and $\Sigma^*$ are uncountable

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 18 views

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GATE2001-2.2 Video Solution
Consider the following statements: $S1:$ There exists infinite sets $A$, $B$, $C$ such that $A \cap (B \cup C)$ is finite. $S2:$ There exists two irrational numbers $x$ and y such that $(x+y)$ is rational. Which of the following is true about $S1$ and $S2$? Only $S1$ is correct Only $S2$ is correct Both $S1$ and $S2$ are correct None of $S1$ and $S2$ is correct

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 7 views

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GATE1993-17 Video Solution
Out of a group of $21$ persons, $9$ eat vegetables, $10$ eat fish and $7$ eat eggs. $5$ persons eat all three. How many persons eat at least two out of the three dishes?

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 18 views

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GATE2008-2 Video Solution
If $P, Q, R$ are subsets of the universal set U, then $(P\cap Q\cap R) \cup (P^c \cap Q \cap R) \cup Q^c \cup R^c$ is $Q^c \cup R^c$ $P \cup Q^c \cup R^c$ $P^c \cup Q^c \cup R^c$ U

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 7 views

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GATE1995-1.20 Video Solution
The number of elements in the power set $P(S)$ of the set $S=\{\{\emptyset\}, 1, \{2, 3\}\}$ is: $2$ $4$ $8$ None of the above

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 7 views

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GATE2000-6 Video Solution
Let $S$ be a set of $n$ elements $\left\{1, 2,....., n\right\}$ and $G$ a graph with 2$^{n}$ vertices, each vertex corresponding to a distinct subset of $S$. Two vertices are adjacent iff the symmetric difference of the corresponding sets has ... Every vertex in $G$ has the same degree. What is the degree of a vertex in $G$? How many connected components does $G$ have?

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 9 views

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GATE2015-2-18 Video Solution
The cardinality of the power set of $\{0, 1, 2, \dots , 10\}$ is _______

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 3 views

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GATE1993-8.3 Video Solution
Let $S$ be an infinite set and $S_1 \dots , S_n$ be sets such that $S_1 \cup S_2 \cup \dots \cup S_n = S$. Then at least one of the set $S_i$ is a finite set not more than one of the set $S_i$ can be finite at least one of the sets $S_i$ is an infinite not more than one of the sets $S_i$ can be infinite None of the above

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 8 views

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GATE2006-IT-23 Video Solution
Let $P$, $Q$ and $R$ be sets let Δ denote the symmetric difference operator defined as $PΔQ=(P \cup Q) - (P ∩ Q).$ Using Venn diagrams, determine which of the following is/are TRUE? $PΔ (Q ∩ R) = (P Δ Q) ∩ (P Δ R)$ $P ∩ (Q ∩ R) = (P ∩ Q) Δ (P Δ R)$ I only II only Neither I nor II Both I and II

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 3 views

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GATE1993-8.4 Video Solution
Let A be a finite set of size n. The number of elements in the power set of $A\times A$ is: $2^{2^n}$ $2^{n^2}$ $(2^n)^2$ $(2^2)^n$ None of the above

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 9 views

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GATE2004-IT-2 Video Solution
In a class of $200$ students, $125$ students have taken Programming Language course, $85$ students have taken Data Structures course, $65$ students have taken Computer Organization course; $50$ students have taken both Programming Language and Data Structures, $35$ ... all the three courses. How many students have not taken any of the three courses? $15$ $20$ $25$ $30$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 10 views

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GATE2006-22 Video Solution
Let $E, F$ and $G$ be finite sets. Let $X = (E ∩ F) - (F ∩ G)$ and $Y = (E - (E ∩ G)) - (E - F)$. Which one of the following is true? $X ⊂ Y$ $X ⊃ Y$ $X = Y$ $X - Y ≠ \phi$ and $Y - X ≠ \phi$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 9 views

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GATE1998-2.4 Video Solution
In a room containing $28$ people, there are $18$ people who speak English, $15$, people who speak Hindi and $22$ people who speak Kannada. $9$ persons speak both English and Hindi, $11$ persons speak both Hindi and Kannada whereas $13$ persons speak both Kannada and English. How many speak all three languages? $9$ $8$ $7$ $6$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 9 views

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GATE2006-IT-24 Video Solution
What is the cardinality of the set of integers $X$ defined below? $X=\{n \mid 1 \leq n ≤ 123, n$ is not divisible by either $2$, $3$ or $5\}$ $28$ $33$ $37$ $44$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 8 views

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GATE2005-8 Video Solution
Let $A, B$ and $C$ be non-empty sets and let $X = ( A - B ) - C$ and $Y = ( A - C ) - ( B - C ).$ Which one of the following is TRUE? $X = Y$ $X ⊂ Y$ $Y ⊂ X$ None of these

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 3 views

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GATE1994-2.4 Video Solution
The number of subsets $\left\{ 1,2, \dots, n\right\}$ with odd cardinality is ___________

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 10 views

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GATE1996-1.1 Video Solution
Let $A$ and $B$ be sets and let $A^c$ and $B^c$ denote the complements of the sets $A$ and $B$. The set $(A-B) \cup (B-A) \cup (A \cap B)$ is equal to $A \cup B$ $A^c \cup B^c$ $A \cap B$ $A^c \cap B^c$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 5 views

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GATE1994-3.8 Video Solution
Give a relational algebra expression using only the minimum number of operators from $(∪, −)$ which is equivalent to $R$ $∩$ $S.$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 13 views

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GATE1994-3.9 Video Solution
Every subset of a countable set is countable. State whether the above statement is true or false with reason.

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 6 views

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GATE2001-3 Video Solution
Prove that powerset $(A \cap B) = \text{powerset}(A) \cap \text{powerset}(B)$ Let $sum(n) = 0 + 1 + 2 + ..... + n$ for all natural numbers n. Give an induction proof to show that the following equation is true for all natural numbers $m$ and $n$: $sum(m+n) = sum(m) + sum(n) + mn$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 3 views

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GATE1995-25b Video Solution
Determine the number of positive integers $(\leq 720)$ which are not divisible by any of $2,3$ or $5.$

asked Apr 18, 2020 in Set Theory & Algebra by admin (573 points) | 3 views

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Applied course test series : Cyclic group
$\{a^2,a^3,a^5,a^{13}\}$ $\{a^2,a^5,a^7,a^{13}\}$ $\{a^2,a^6,a^{10},a^{14}\}$ None of these How to solve the above question?

asked Feb 2, 2020 in Set Theory & Algebra by vishal burnwal (77 points) | 176 views

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Ace test series : Sets

asked Jan 12, 2020 in Set Theory & Algebra by Chirag Shilwant (191 points) | 54 views

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