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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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Recent questions in Set Theory & Algebra

#341
9.6k
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5 answers
16 votes
GATE CSE 2020 | Question: 17
Let $\mathcal{R}$ be the set of all binary relations on the set $\{1,2,3\}$. Suppose a relation is chosen from $\mathcal{R}$ at random. The probability that the chosen relation is reflexive (round off to $3$ decimal places) is ______.
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#342
9.6k
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4 answers
15 votes
GATE CSE 2020 | Question: 18
Let $G$ be a group of $35$ elements. Then the largest possible size of a subgroup of $G$ other than $G$ itself is _______.
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#343
1.4k
views
1 answers
3 votes
ISRO2020-76
If $A=\{x,y,z\}$ and $B=\{u,v,w,x\}, $ and the universe is $\{s,t,u,v,w,x,y,z\}$. Then $(A \cup \overline{B}) \cap (A \cap B)$ is equal to$\{u,v,w,x\}$\{ \: \}$\{u,v,w,x,y,z\}$\{u,v,w\}$
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#344
1.3k
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1 answers
1 votes
ISI2014-DCG-5
Consider the sets defined by the real solutions of the inequalities$A = \{(x,y):x^2+y^4 \leq 1\} \:\:\:\:\:\:\: B=\{(x,y):x^4+y^6 \leq 1\}$ ... Each of the sets $A - B, \: B - A$ and $A \cap B$ is non-emptynone of the above
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#345
520
views
1 answers
2 votes
ISI2014-DCG-15
Let $\mathbb{N}=\{1,2,3, \dots\}$ be the set of natural numbers. For each $n \in \mathbb{N}$, define $A_n=\{(n+1)k, \: k \in \mathbb{N} \}$. Then $A_1 \cap A_2$ equals$A_3$A_4$A_5$A_6$
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#346
1.2k
views
2 answers
1 votes
ISI2014-DCG-35
Let $A$ and $B$ be disjoint sets containing $m$ and $n$ elements respectively, and let $C=A \cup B$. Then the number of subsets $S$ (of $C$) which contains $p$ ...
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#347
880
views
2 answers
1 votes
ISI2015-MMA-5
A set contains $2n+1$ elements. The number of subsets of the set which contain at most $n$ elements is$2^n$2^{n+1}$2^{n-1}$2^{2n}$
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#348
1.1k
views
2 answers
2 votes
ISI2015-MMA-7
Let $X$ be the set $\{1, 2, 3, 4, 5, 6, 7, 8, 9, 10 \}$. Define the set $\mathcal{R}$ by $\mathcal{R} = \{(x,y) \in X \times X : x$ ... $.Then the number of elements in $\mathcal{R}$ is$40$36$34$33$
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#349
790
views
1 answers
0 votes
ISI2015-MMA-23
Let $X$ be a nonempty set and let $\mathcal{P}(X)$ denote the collection of all subsets of $X$. Define $f: X \times \mathcal{P}(X) \to \mathbb{R}$ ... ,A) \cdot f(x,B)$f(x,A)\:+ \mid f(x,A)\: - f(x,B) \mid $
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#350
564
views
1 answers
0 votes
ISI2015-MMA-31
Consider the sets defined by the real solutions of the inequalities$A = \{(x,y):x^2+y^4 \leq 1 \} \:\:\:\:\:\:\:\: B = \{ (x,y):x^4+y^6 \leq 1\}$ ... $A - B, \: B - A$ and $A \cap B$ is non-emptynone of the above
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#351
1.6k
views
3 answers
0 votes
ISI2015-MMA-92
Consider the group $G=\begin{Bmatrix} \begin{pmatrix} a & b \\ 0 & a^{-1} \end{pmatrix} : a,b \in \mathbb{R}, \: a>0 \end{Bmatrix}$ ... the quotient group is isomorphic to $\mathbb{R}^+$ (the group of positive reals with multiplication).
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#352
945
views
1 answers
3 votes
ISI2015-MMA-93
Let $G$ be a group with identity element $e$. If $x$ and $y$ are elements in $G$ satisfying $x^5y^3=x^8y^5=e$, then which of the following conditions is true?$x=e, \: y=e$x\neq e, \: y=e$x=e, \: y \neq e$x\neq e, \: y \neq e$
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#353
847
views
1 answers
0 votes
ISI2015-MMA-94
Let $G$ be the group $\{\pm1, \pm i \}$ with multiplication of complex numbers as composition. Let $H$ be the quotient group $\mathbb{Z}/4 \mathbb{Z}$. Then the number of nontrivial group homomorphisms from $H$ to $G$ is$4$1$2$3$
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#354
394
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2 answers
0 votes
ISI2015-DCG-17
The set $\{(x,y): \mid x \mid + \mid y \mid \leq 1\}$ is represented by the shaded region in
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#355
432
views
1 answers
1 votes
ISI2015-DCG-27
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ ... area of triangle $x,$ isone-one and intoone-one and ontomany-one and ontomany-one and into
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#356
327
views
1 answers
0 votes
ISI2015-DCG-35
Let $A$, $B$ and $C$ ... $(2)$ is not$(2)$ is correct but $(1)$ is notBoth $(1)$ and $(2)$ are incorrect
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#357
503
views
1 answers
3 votes
ISI2015-DCG-36
Suppose $X$ and $Y$ are finite sets, each with cardinality $n$. The number of bijective functions from $X$ to $Y$ is$n^n$n \log_2 n$n^2$n!$
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#358
322
views
0 answers
0 votes
ISI2015-DCG-37
Suppose $f_{\alpha} : [0,1] \to [0,1],\:\: -1 < \alpha < \infty$ ... one-one and onto) functionA surjective (onto ) functionAn injective (one-one) functionWe cannot conclude about the type
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#359
509
views
2 answers
1 votes
ISI2015-DCG-49
The domain of the function $\text{ln}(3x^2-4x+5)$ isset of positive real numbersset of real numbersset of negative real numbersset of real numbers larger than $5$
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#360
339
views
1 answers
0 votes
ISI2016-DCG-27
If $A$ be the set of triangles in a plane and $R^{+}$ be the set of all positive real numbers, then the function $f\::\:A\rightarrow R^{+},$ ... area of triangle $x,$ isone-one and intoone-one and ontomany-one and ontomany-one and into
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