Consider the following sets, where n ≥ 2:S1 : Set of all n × n. matrices with entries from the set {a, b, c}S2 : Set of all functions from the set {0, 1, 2, …, n 2 – 1} to the set {0,1,2}Which of the following choice(s) is/are correct? [MSQ || GATE 2021 Set-2 || 1 Marks ]

There exists a surjection from S1 to S2 .

There does not exist an injection from S1 to S2 .

There does not exist a bijection from S1 to S2 .

Let G be a group of order 6, and H be subgroup of G such that 1 < |H| < 6.Which one of the following options is correct? [MCQ || GATE 2021 Set-1 || 2 Marks ]

G may not be cyclic, but H is always cyclic.

G is always cyclic, but H may not be cyclic.

Both G and H are always cyclic.

Both G and H may not be cyclic.

Set theory | GATE PYQ [2010-2025]

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Question 1

Consider the following sets, where n ≥ 2:

S₁ : Set of all n × n. matrices with entries from the set {a, b, c}

S₂ : Set of all functions from the set {0, 1, 2, …, n 2 – 1} to the set {0,1,2}

Which of the following choice(s) is/are correct? [MSQ || GATE 2021 Set-2 || 1 Marks ]

There exists a bijection from S₁ to S₂ .
There exists a surjection from S₁ to S₂ .
There does not exist an injection from S₁ to S₂ .
There does not exist a bijection from S₁ to S₂ .

Question 2

The number of integers between 1 and 500 (both inclusive) that are divisible by 3 or 5 or 7 is ____________. [ NAT || GATE 2017 Set-1 || 1 Marks ]

Question 3

Consider the set X = {a,b,c,d e} under the partial ordering

R = {(a,a),(a,b),(a,c),(a,d),(a,e),(b,b),(b,c),(b,e),(c,c),(c,e),(d,d),(d,e),(e,e)}.

The Hasse diagram of the partial order (X,R) is shown below.

The minimum number of ordered pairs that need to be added to R to make (X, R) a lattice is _________. [ NAT || GATE 2017 Set-2 || 1 Marks ]

Question 4

Let G be a finite group on 84 elements. The size of a largest possible proper subgroup of G is _________. [NAT || GATE 2018 || 1 Marks ]

Question 5

Let N be the set of natural numbers.

Consider the following sets,

P: Set of Rational numbers (positive and negative)

Q: Set of functions from {0, 1} to N

R: Set of functions from N to {0, 1}

S: Set of finite subsets of N.

Which of the above sets are countable? [MCQ || GATE 2018 || 2 Marks ]

Q and S only
P and S only
P and R only
P, Q and S only

Question 6

Let G be an arbitrary group. Consider the following relations on G:

R₁ : ∀a,b ∈ G, aR₁ b if and only if ∃g ∈ G such that a = g^-1 bg

R₂ : ∀a,b ∈ G, aR₂b if and only if a = b ^-1

Which of the above is/are equivalence relation/relations? [MCQ || GATE 2019 || 1 Marks ]

R₂ only
R₁ and R₂
Neither R₁ nor R₂
R₁ only

Question 7

Let Σ be the set of all bijections from {1,..., 5} to {1, ..., 5}, where id denotes the identity function i.e., id(j) = j, ∀j. Let º denote composition on functions. For a string x = x ₁ x₂ ... x _n ∈ Σ ⁿ , n ≥ 0. Let π(x) = x ₁ º x ₂ º ... º x _n .

Consider the language L = {x ∈ Σ* | π(x) = id}. The minimum number of states in any DFA accepting L is ______. [NAT || GATE 2019 || 2 Marks ]

Question 8

Let G be a group of 35 elements. Then the largest possible size of a subgroup of G other than G itself is ______. [NAT || GATE 2020 || 1 Marks ]

Question 9

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 is reflexive (round off to 3 decimal places) is _____. [NAT || GATE 2020 || 1 Marks ]

0.125

Question 10

Let G be a group of order 6, and H be subgroup of G such that 1 < |H| < 6.

Which one of the following options is correct? [MCQ || GATE 2021 Set-1 || 2 Marks ]

G is always cyclic, but H may not be cyclic.
G may not be cyclic, but H is always cyclic.
Both G and H are always cyclic.
Both G and H may not be cyclic.

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Discrete Mathematics

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