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

$$\scriptsize{\overset{{\large{\textbf{Mark Distribution in Previous GATE}}}}{\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|c|c|c|}\hline \textbf{Year}& \textbf{2026 - 1}& \textbf{2026 - 2}& \textbf{2025 - 1}& \textbf{2025 - 2}& \textbf{2024 - 1}& \textbf{2024 - 2}& \textbf{2023}& \textbf{2022}& \textbf{2021 - 1}& \textbf{2021 - 2}&\textbf{Minimum}&\textbf{Average}&\textbf{Maximum}\\\hline \textbf{1 Mark Count}&0&1&1&0&1&1&0&1&0&1&0&0.6&1\\\hline \textbf{2 Marks Count}&0&0&1&1&1&1&2&0&2&1&0&0.9&2\\\hline \textbf{Total Marks}&0&1&3&2&3&3&4&1&4&3&\bf{0}&\bf{2.4}&\bf{4}\\\hline \end{array}}}$$

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Recent questions in Set Theory & Algebra

4 4 votes
1 1 answer
389
389 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $A$ be the set of all finite binary strings, including the empty string $\epsilon$. Define $F:A\to A$ as follows: $F(w)$ is obtained by writing $0$ before the string ...
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1 1 vote
1 1 answer
262
262 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $A$ and $B$ be non-empty sets and let $f:A\to B$ be a function. For $S\subseteq A$, define $S^c=A-S$. For $Y\subseteq B$, define $Y^c=B-Y$. Which of the following sta...
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2 2 votes
1 1 answer
246
246 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Find the domain of the function $F(x)=\sqrt{\ln(x-1)}+\ln\left(\dfrac{\sqrt{x+1}-2}{x^2-5x+6}\right)$.$[2,\infty)$ $(2,\infty)$ $(2,3)\cup(3,\infty)$ $[2,3)\cup(3,\infty)...
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1 1 vote
2 2 answers
237
237 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $A=\{1,2,3,4,5,6\}$. Find the number of bijections $f:A\to A$ such that $f(1)\neq1$, $f(2)\neq2$, $f(3)\neq3$, and $f(4)=5$ if and only if $f(5)=4$.
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4 4 votes
1 1 answer
228
228 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $A$ be the set of all non-constant linear functions from $\mathbb R$ to $\mathbb R$. Define $T:A\to\mathbb R$ by $T(\phi)=$ the unique real number $x$ for which $\phi...
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0 0 votes
2 2 answers
135
135 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $F$ and $G$ be Boolean functions of degree $n$, where the order relation is defined by $F\leq G$ if and only if $F(x)\leq G(x)$ for every $x\in{0,1}^n$. Also define $...
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0 0 votes
1 1 answer
88
88 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
A $4$-ary boolean function is a function $f:\{0,1\}^4\to\{0,1\}$. Let $h$ be a fixed $4$-ary boolean function. We say that another $4$-ary boolean function $g$ is strongl...
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0 0 votes
1 1 answer
100
100 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $a_n$ denote the number of strictly increasing sequences of positive integers whose first term is $1$, whose last term is $n$, and in which no two consecutive terms d...
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0 0 votes
1 1 answer
84
84 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $F:\mathbb{Z}_{\geq 0}\times \mathbb{Z}_{\geq 0}\to \mathbb{Z}_{\geq 0}$ be defined by $F(m,n)=\frac{(m+n)(m+n+1)}{2}+n$. Which of the following statements is correct...
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0 0 votes
1 1 answer
113
113 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $f:A\to B$ and $g:B\to C$ be functions, and let $g\circ f:A\to C$ be defined by $(g\circ f)(x)=g(f(x))$. Which of the following statements is/are always TRUE?If $g\ci...
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0 0 votes
1 1 answer
108
108 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $A=\mathbb{R}-\{1\}$ and $B=\mathbb{R}-\{2\}$. Define $f:A\to B$ by $f(x)=\frac{2x+1}{x-1}$ and $g:B\to A$ by $g(x)=\frac{x+1}{x-2}$. Which of the following statement...
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0 0 votes
1 1 answer
104
104 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $f:\mathbb{R}\to\mathbb{R}$ and $g:\mathbb{R}\to\mathbb{R}$ be defined by $f(x)=x^3-2x+1$ and $g(x)=\sqrt[3]{x-1}$. Which of the following is TRUE?$f$ and $g$ are inv...
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1 1 vote
1 1 answer
98
98 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $A$ be a finite set with $m$ elements and let $B$ be a finite set with $n$ elements, where $m\ge n$. How many functions $f:A\to B$ are onto?$n^m$ $m^n$ $n^m-\binom{n}...
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1 1 vote
1 1 answer
94
94 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $f:A\to B$ be a function. Which of the following statements is/are TRUE?If $f$ is one-to-one, then $f$ must be invertible. If $f$ is onto, then $f$ must be invertible...
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2 2 votes
1 1 answer
154
154 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $f:X\to Y$ be a one-to-one function, and let $A,B\subseteq X$. Then $f(A\cap B)$ is always equal to:$f(A)\cap f(B)$ $f(A)\cup f(B)$ $f(A)-f(B)$ $f(B)-f(A)$
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1 1 vote
1 1 answer
125
125 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $A$ be a set with $n$ elements, where $n\ge 1$. How many functions $f:A\to A$ satisfy $f(f(x))=f(x)$ for every $x\in A$?$n^n$ $\sum_{k=1}^{n}\binom{n}{k}k^{n-k}$ $n!$...
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1 1 vote
1 1 answer
167
167 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $f:\mathbb{R}\to\mathbb{R}$ be a bounded uniformly continuous function, and let $g:\mathbb{R}\to\mathbb{R}$ be continuous. Then $g\circ f$ is:Uniformly continuous Con...
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1 1 vote
1 1 answer
111
111 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $f$ be a one-to-one function from $[0,1]$ into $[0,1]$. No continuity is assumed. Which of the following must be true?$f$ must be onto $[0,1]$ The range of $f$ must c...
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1 1 vote
1 1 answer
114
114 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
If $f(x)=\frac{1}{x}$ and $g(x)=\frac{x-1}{x+1}$, then $\frac{f(g(x))+g(f(x))}{f(g(x))-g(f(x))}$ is:$\frac{2}{x+f(x)}$ $\frac{x+f(x)}{2}$ $\frac{g(x)}{f(x)}$ $\frac{f(x)}...
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1 1 vote
2 2 answers
137
137 views
GO Classes DPP | GATE CS | Discrete Mathematics| Functions
Let $g:A\to B$ and $f:B\to C$, where $A$, $B$, and $C$ are finite sets with $|A|=|C|=6$ and $|B|=8$. Suppose $f\circ g:A\to C$ is bijective. Which of the following statem...
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