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Syllabus: Sets, Relations, Functions, Partial orders, Lattices, Monoids, Groups.

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|}\hline
\textbf{Year}&\textbf{2024-1} &\textbf{2024-2} &\textbf{2023} & \textbf{2022} & \textbf{2021-1}&\textbf{2021-2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}
\\\hline\textbf{1 Mark Count} &1&1&0& 1&0&1&0&0.83&1
\\\hline\textbf{2 Marks Count} &1&1&2& 0 &2&1&0&1.16&2
\\\hline\textbf{Total Marks} & 3&3&4&1&4&3&\bf{1}&\bf{3}&\bf{4}\\\hline
\end{array}}}$$

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#221
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Computer Architecture and Organization
A device with a data transfer rate of 20 KB/sec is connected to a CPU. Data is transferred byte-wise. Let the interrupt overhead be 10 ... of operating the device under interrupt mode over operating it under program-controlled mode?
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#222
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2 votes
Applied Grand Test 2
Consider the equivalence relation R induced by the partition P={{1},{3},{2,4,5,6}} of set A={1,2,3,4,5,6}.The number of ordered pairs in R is ____
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#223
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Applied gate testseries: No of abelian groups of order 160 is
Can somebody explain the following question and answer?
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#224
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TIFR CSE 2021 | Part A | Question: 5
Let $n, m$ and $k$ be three positive integers such that $n \geq m \geq k$. Let $S$ be a subset of $\left \{ 1, 2,\dots, n \right \}$ of size $k$. Consider sampling a ... 1-\frac{k!\binom{n}{k}}{n^{k}}$1-\frac{k!\binom{n}{k}}{m^{k}}$
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#225
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TIFR CSE 2021 | Part A | Question: 9
Fix $n\geq 6$. Consider the set $\mathcal{C}$ of binary strings $x_{1}, x_{2} \dots x_{n}$ of length $n$ such that the bits satisfy the following set of equalities, ... $3$ then $\left | \mathcal{C} \right |=1$.
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#226
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GATE CSE 2021 Set 2 | Question: 11
Consider the following sets, where $n \geq 2$:$S_1$: Set of all $n \times n$ matrices with entries from the set $\{ a, b, c\}$ ... a bijection from $S_1$ to $S_2$There does not exist an injection from $S_1$ to $S_2$
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#227
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GATE CSE 2021 Set 2 | Question: 37
For two $n$-dimensional real vectors $P$ and $Q$, the operation $s(P,Q)$ ... )=0$. What is the maximum cardinality possible for the set $\mathcal{L}$?$9$10$11$100$
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#228
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GATE CSE 2021 Set 1 | Question: 34
Let $G$ be a group of order $6$, and $H$ be a subgroup of $G$ such that $1<|H|<6$. Which one of the following options is correct?Both $G$ and $H$ are ... $G$ is always cyclic, but $H$ may not be cyclicBoth $G$ and $H$ may not be cyclic
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#229
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GATE CSE 2021 Set 1 | Question: 43
A relation $R$ is said to be circular if $a\text{R}b$ and $b\text{R}c$ together imply $c\text{R}a$.Which of the following options is/are correct ... relation.If a relation $S$ is transitive and circular, then $S$ is an equivalence relation.
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#230
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UGC NET CSE | October 2020 | Part 2 | Question: 1
The number of positive integers not exceeding $100$ that are either odd or the square of an integer is _______$63$59$55$50$
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#231
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Kenneth Rosen Edition 7 Exercise 2.5 Question 40 (Page No. 177)
Show that if $S$ is a set, then there does not exist an onto function $f$ from $S$ to $P(S),$ the power set of $S$ ... $s$ can exist for which $f (s) = T.]$
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#232
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Kenneth Rosen Edition 7 Exercise 2.5 Question 39 (Page No. 177)
We say that a function is computable if there is a computer program that finds the values of this function. Use question $37$ and $38$ to show that there are functions that are not computable.
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#233
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Kenneth Rosen Edition 7 Exercise 2.5 Question 38 (Page No. 177)
Show that the set of functions from the positive integers to the set $\{0, 1, 2, 3, 4, 5, 6, 7, 8, 9\}$ is uncountable. [Hint: First set up a one-to-one correspondence ... $f$ with $f (n) = dn.]$
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#234
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Kenneth Rosen Edition 7 Exercise 2.5 Question 37 (Page No. 177)
Show that the set of all computer programs in a particular programming language is countable. [Hint: A computer program written in a programming language can be thought of as a string of symbols from a finite alphabet.]
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#235
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Kenneth Rosen Edition 7 Exercise 2.5 Question 36 (Page No. 177)
Show that there is a one-to-one correspondence from the set of subsets of the positive integers to the set real numbers between $0$ and $1$. Use this result and ... $35.]$
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#236
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Kenneth Rosen Edition 7 Exercise 2.5 Question 35 (Page No. 177)
Show that there is no one-to-one correspondence from the set of positive integers to the power set of the set of positive integers. [Hint: Assume that there is ... in the list. Show that this new bit string cannot appear in the list.]
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#237
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Kenneth Rosen Edition 7 Exercise 2.5 Question 34 (Page No. 177)
Show that $(0, 1)$ and $R$ have the same cardinality. [Hint: Use the Schröder-Bernstein theorem.]
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#238
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Kenneth Rosen Edition 7 Exercise 2.5 Question 33 (Page No. 177)
Use the Schröder-Bernstein theorem to show that $(0, 1)$ and $[0, 1]$ have the same cardinality.
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#239
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Kenneth Rosen Edition 7 Exercise 2.5 Question 32 (Page No. 177)
Show that when you substitute $(3n + 1)^{2}$ for each occurrence of $n$ and $(3m + 1)^{2}$ for each occurrence of m in the right-hand side of ... open question whether there is a one-to-one polynomial function $Q \times Q \rightarrow Q.$
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#240
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Kenneth Rosen Edition 7 Exercise 2.5 Question 31 (Page No. 177)
Show that $Z^{+} \times Z^{+}$ is countable by showing that the polynomial function $f : Z^{+} \times Z^{+}\rightarrow Z^{+}$ with $f(m, n) = \dfrac{(m + n − 2)(m + n − 1)}{2} + m$ is one-to one and onto.
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