# Set Theory & Algebra · Discrete Mathematics · GATE CSE

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## Marks 1

GATE CSE 2020
Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______.
GATE CSE 2020
Let R be the set of all binary relations on the set {1,2,3}. Suppose a relation is chosen from R at random. The probability that the chosen relation i...
GATE CSE 2019
Let G be an arbitrary group. Consider the following relations on G : R1: ∀a,b ∈ G, aR1b if and only if ∃g ∈ G such that a = g-1bg R2: ∀a,b ∈ G, aR2b...
GATE CSE 2019
Let U = {1, 2 ,..., n}. Let A = {(x, X) | x ∈ X, X ⊆ U}. Consider the following two statements on |A|. I. |A| = n2n–1 II. |A| = $$\sum\limits_{k = 1}^... GATE CSE 2015 Set 1 For a set A, the power set of A is denoted by 2A. If A = {5, {6}, {7}}, which of the following options are TRUE? I.$$\phi \in {2^A}$$II.$$\phi \s...
GATE CSE 2015 Set 2
Let $$𝑅$$ be the relation on the set of positive integers such that $$aRb$$ if and only if $$𝑎$$ and $$𝑏$$ are distinct and have a common divisor ...
GATE CSE 2015 Set 2
The cardinally of the power set of $$\left\{ {0,1,2,\,\,....,\,\,10} \right.\left. \, \right\}$$ is _____________.
GATE CSE 2015 Set 3
Suppose $$𝑈$$ is the power set of the set $$S = \left\{ {1,2,3,4,5,6,} \right\}$$. For any $$T \in U,$$ let $$\left| T \right|$$ denote the number of...
GATE CSE 2014 Set 3
Let $$X$$ and $$Y$$ be finite sets and $$f:X \to Y$$ be a function. Which one of the following statements is TRUE?
GATE CSE 2013
A Binary operation $$\oplus$$ on a set of integers is defined as $$x$$ $$\oplus$$ $$y$$ $$= {x^2} + {y^2}$$. Which one of the following statem...
GATE CSE 2013
Which one of the following functions is continuous at $$x = 3$$?
GATE CSE 2010
What is the possible number of reflexive relations on a set $$5$$ elements?
GATE CSE 2010
Consider the set $$S = \left\{ {1,\,\omega ,\,{\omega ^2}} \right\},$$ where $$\omega$$ and $${{\omega ^2}}$$, are cube roots of unity. If $$*$$ de...
GATE CSE 2009
Which one of the following in NOT necessarily a property of Group?
GATE CSE 2009
consider the binary relation $$R = \left\{ {\left( {x,y} \right),\,\left( {x,z} \right),\,\left( {z,x} \right),\,\left( {z,y} \right)} \right\}$$ on t...
GATE CSE 2008
If $$P, Q, R$$ are subsets of the universal set $$U$$, then $$\left( {P \cap Q \cap R} \right) \cup \left( {{P^c} \cap Q \cap R} \right) \cup {Q^c} \... GATE CSE 2007 Let$$S$$be a set6 of$$n$$elements. The number of ordered pairs in the largest and the smallest equivalence relations on$$S$$are GATE CSE 2007 What is the maximum number of different Boolean functions involving$$n$$Boolean variables? GATE CSE 2006 Let$$X,. Y, Z$$be sets of sizes$$x, y$$and$$z$$respectively. Let$$W = X x Y$$and$$E$$be the set of all subjects of$$W$$. The number ... GATE CSE 2006 The set$$\left\{ {1,\,\,2,\,\,3,\,\,5,\,\,7,\,\,8,\,\,9} \right\}$$under multiplication modulo 10 is not a group. Given below are four plausible rea... GATE CSE 2006 A relation$$R$$is defined on ordered pairs of integers as follows:$$\left( {x,y} \right)R\left( {u,v} \right)\,if\,x < u$$and$$y > v$$. Th... GATE CSE 2006 For the set$$N$$of natural numbers and a binary operation$$f:N \times N \to N$$, an element$$z \in N$$is called an identity for$$f$$if$$f\left...
GATE CSE 2005
The set $$\left\{ {1,\,\,2,\,\,4,\,\,7,\,\,8,\,\,11,\,\,13,\,\,14} \right\}$$ is a group under multiplication modulo $$15$$. The inverse of $$4$$ and ...
GATE CSE 2005
Let $$f$$ be a function from a set $$A$$ to a set $$B$$, $$g$$ a function from $$B$$ to $$C$$, and $$h$$ a function from $$A$$ to $$C$$, such that $$h... GATE CSE 2005 Let$$A$$,$$B$$and$$C$$be non-empty sets and let$$X = (A - B) - C$$and$$Y = (A - C) - (B - C)$$Which one of the following is TRUE? GATE CSE 2005 The following is the Hasse diagram of the poset$$\left[ {\left\{ {a,b,c,d,e} \right\}, \prec } \right]$$The poset is: ... GATE CSE 2004 Consider the binary relation:$$S = \left\{ {\left( {x,y} \right)|y = x + 1\,\,and\,\,x,y \in \left\{ {0,1,2,...} \right\}} \right\}$$The reflexive ... GATE CSE 2004 The number of different$$nxn$$symmetric matrices with each elements being either$$0$$or$$1$$is (Note: power ($$2,x$$) is s... GATE CSE 2004 Let$${R_1}$$be a relation from$$A = \left\{ {1,3,5,7} \right\}$$to$$B = \left\{ {2,4,6,8} \right\}$$and$${R_2}$$be another relation from$$B$$... GATE CSE 2001 Consider the following relations:$${R_1}\,\,\left( {a,\,\,b} \right)\,\,\,iff\,\,\left( {a + b} \right)$$is even over the set of integers$${R_2}\,\...
GATE CSE 1999
The number of binary relations on a set with $$n$$ elements is:
GATE CSE 1998
The number of functions from an $$m$$ element set to an $$n$$ element set is
GATE CSE 1998
Let $${R_1}$$ and $${R_2}$$ be two equivalence relations on a set. Consider the following assertions: (i)$$\,\,\,\,{R_1} \cup {R_2}$$ is an euivalence...
GATE CSE 1998
Suppose $$A$$ is a finite set with $$n$$ elements. The number of elements in the Largest equivalence relation of $$A$$ is
GATE CSE 1997
The number of equivalence relations on the set $$\left\{ {1,2,3,4} \right\}$$ is
GATE CSE 1996
Let $$A$$ and $$B$$ be sets and let $${A^c}$$ and $${B^c}$$ denote the complements of the sets $$A$$ and $$B$$. The set $$\left( {A - B} \right) \cup ... GATE CSE 1996 Let$$XX = \left\{ {2,3,6,12,24} \right\}$$. Let$$ \le $$the partial order defined by$$x \le y$$if$$x$$divides$$y$$. The number of edges... GATE CSE 1996 Suppose$$X$$and$$Y$$are sets and$$\left| X \right|$$and$$\left| Y \right|$$are their respective cardinalities. It is given that there are exac... GATE CSE 1996 Which of the following statements is false? GATE CSE 1995 The number of elements in the power set$$P(S)$$of the set$$S = \left\{ {\left\{ \phi \right\},1,\left\{ {2,3} \right\}} \right\}$$is GATE CSE 1995 Let$$R$$be a symmetric and transitive relation on a set$$A$$. Then GATE CSE 1993 Let$$S$$be an infinite set and$${S_1},\,\,{S_2},....\,\,{S_n}$$be sets such that$${S_1} \cup {S_2} \cup ....... \cup {S_n} = S$$. Then GATE CSE 1993 Let$${\rm A}$$be a finite set of size$$n$$. The number of elements in the power set of$${\rm A} \times {\rm A}$$is GATE CSE 1987 State whether the following statement are TRUE or FALSE: (a) The union of two equivalence relations is also an equivalence relation. GATE CSE 1987 (a) How many binary relations are there on a set A with n elements? (b) How many one - to - one functions are there from a set A with n elements onto ... ## Marks 2 GATE CSE 2023 Let X be a set and 2$$^X$$denote the powerset of X. Define a binary operation$$\Delta$$on 2$$^X$$as follows:$$A\Delta B=(A-B)\cup(B-A)$$. Let$$H...
GATE CSE 2021 Set 1
A relation R is said to be circular if aRb and bRc together imply cRa. Which of the following options is/are correct?...
GATE CSE 2019
Consider the first order predicate formula φ: ∀x[(∀z z|x ⇒ ((z = x) ∨ (z = 1))) ⇒ ∃w (w > x) ∧ (∀z z|w ⇒ ((w = z) ∨ (z = 1)))] Here 'a|b' denotes t...
GATE CSE 2018
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