Gujarat Technological UniversityWinter 2023 examination

GTU 3140708 Discrete Mathematics (DM) Winter 2023 Question Paper & PDF

B.E. · Computer Engineering · Semester 4 · Subject code 3140708
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Total
70 marks
To pass
23 marks (33%)
Duration
2.5 Hours 10:30 AM – 1:00 PM
Questions
5 with internal choice
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14 of 25 questions are types of question GTU sets again with new data. Each is marked below with the years it appeared.See all 23 repeated 3140708 questions, most asked first

Q.1

14 Marks
(a)
3 marksSame type asked in 3 of 5 papers
A function 𝑓: ℝ+ → ℝ is defined by 𝑓(𝑥) = 𝑥² − 8. Check whether 𝑓 is
one-one and onto.

Also asked in Summer 2026 Q3a, Winter 2024 Q1a

(b)
4 marksSame type asked in 3 of 5 papers

Determine the relation ∥ (parallel) on the set L of lines in the plane are reflexive, symmetric, anti-symmetric, transitive, irreflexive.

Also asked in Winter 2024 Q1b, Summer 2024 Q1b

(c)
7 marks
(
i)Show that the functions 𝑓(𝑥) = 𝑥³ and 𝑔(𝑥) = 𝑥¹ 3 for 𝑥 ∈ ℝ are inverses of one another.
ii)Check whether the following graphs are isomorphic.3 Marks

Q.2

14 Marks
(a)
3 marksSame type asked in 4 of 5 papers
Identify the statement (𝑝⋀𝑞) ⋀ ¬(𝑝⋁ 𝑞) is tautology or contradiction.

Also asked in Winter 2025 Q3a, Winter 2025 Q3b, Winter 2024 Q2b, Summer 2024 Q2a

(b)
4 marks
(
i)Show that in a group (𝐺,∗), for any 𝑎, 𝑏 ∈ 𝐺 if (𝑎 ∗ 𝑏)² = 𝑎² ∗ 𝑏² then (𝐺,∗) must be abelian.
ii)If (𝐺,∗) be a group then for any two elements 𝑎 and 𝑏 of (𝐺,∗), prove that (𝑎 ∗ 𝑏)⁻¹ = 𝑏⁻¹ ∗ 𝑎⁻¹.
(c)
7 marksSame type asked in 3 of 5 papers
Use a truth table to determine whether the following argument form is
valid.
𝑝 → 𝑞
𝑞 → 𝑟
∴ 𝑝 → 𝑟

Also asked in Winter 2024 Q2c, Summer 2024 Q2c OR

(c)
7 marksSame type asked in 3 of 5 papers

Express the following using predicate, quantifier and logical connectives. Also verify the validity of the consequence.

i)Every computer science major takes discrete mathematics. Natasha is taking discrete mathematics. Therefore, Natasha is a computer science major.
ii)All parrots like fruit. My pet bird is not a parrot. Therefore, my pet bird does not like fruit.

Also asked in Winter 2024 Q2c OR, Summer 2024 Q2c

Q.3

14 Marks
(a)
3 marks

Show that (ℤ5 ∗ ,×6) is cyclic group, where ℤ5 ∗ = ℤ5\{0}.

(b)
4 marks

Given A = {x ∶ x is an integer and 1 ≤ x ≤ 5}, 𝐵 = {3, 4, 5, 17}, and 𝐶 = {1, 2, 3, . . . }, find 𝐴 ∩ 𝐵, 𝐴 ∪ 𝐵, 𝐴 ∩ 𝐵 ∩ 𝐶 and 𝐴 ∪ 𝐶.

(c)
7 marks
(
i)Show that every subgroup of a cyclic group is normal
ii)The subset 𝐻 = {0,2} is a subgroup of (ℤ4, +4). Find all left and right cosets of 𝐻 in (ℤ4, +4). Is 𝐻 a normal subgroup?3 Marks
(a)
3 marksSame type asked in 2 of 5 papers
Define 𝑓: (ℕ × ℕ,∗) → (ℚ,×) by 𝑓(𝑎, 𝑏) = 𝑎
𝑏. Show that 𝑓 is a
homomorphism.

Also asked in Winter 2024 Q3a

(b)
4 marksSame type asked in 2 of 5 papers

In a class of 50 students, 12 enrolled for both Mathematics and Science, 32 enrolled for Science. If the students of the class enrolled for at least one of the two subjects, then how many students enrolled for only Mathematics but not Science?

Also asked in Winter 2024 Q3b

(c)
7 marksSame type asked in 2 of 5 papers

Consider the ring ℤ10 = {0, 1, 2, … ,9} of integers modulo 10.

aFind the units of ℤ10.
bFind −3, −8 and 3−1.
cLet 𝑓 (𝑥) = 2𝑥² + 4𝑥 + 4. Find the roots of 𝑓 (𝑥) over ℤ10.

Also asked in Winter 2024 Q3c OR

Q.4

14 Marks
(a)
3 marks

Let 𝑋 = {1, 2, 3, 4} and 𝑅 = {〈𝑥, 𝑦〉/ 𝑥 > 𝑦} . Draw the graph of 𝑅 and also give its matrix.

(b)
4 marks

Find out maximal compatibility blocks of following simplified graph of digraph and write its relation matrix.

(c)
7 marks
Prove that 〈𝑆30, 𝐷〉 is a Boolean algebra.
(a)
3 marksSame type asked in 2 of 5 papers

Define equivalence relation. Let 𝑋 = {1, 2, 3, 4,5}, 𝑅 = {〈𝑥, 𝑦〉/ 𝑥 is divisible by 𝑦}. Check whether the relation an equivalence relation?

Also asked in Winter 2024 Q4a

(b)
4 marks

Let 𝐴 = {𝑎, 𝑏, 𝑐, 𝑑} and 𝜌(𝐴) its power set. Let ⊆ be the inclusion relation on the elements of 𝜌(𝐴). Draw the Hasse diagram of 〈𝜌(𝐴), ⊆〉.

(c)
7 marksSame type asked in 4 of 5 papers

Solve 𝑎𝑛 = 11𝑎𝑛−1 − 39𝑎𝑛−2 + 45𝑎𝑛−3, 𝑎0 = 5, 𝑎1 = 11, 𝑎2 = 25.

Also asked in Winter 2025 Q3c OR, Winter 2024 Q4b OR, Summer 2024 Q4b OR

Q.5

14 Marks
(a)
3 marksSame type asked in 2 of 5 papers

A tree 𝑇 has 3 vertices of degree 4, 3 vertices of degree 3. Find the number of pendant vertices in tree 𝑇.

Also asked in Winter 2024 Q5a OR

(b)
4 marksSame type asked in 2 of 5 papers

Find reachable set of each node of the given digraph. Also find 𝑑(𝑉1, 𝑉3), 𝑑(𝑉3, 𝑉1).

Also asked in Winter 2024 Q5b OR

(c)
7 marks

Define Strong, unilateral, week component. Also Find strong, unilateral, week component from the given digraph.

(a)
3 marks
Define null graph, centre of a graph and complete graph.
(b)
4 marksSame type asked in 4 of 5 papers
Find the path matrix using adjacency matrix of the following graph.

Also asked in Summer 2026 Q3c OR, Winter 2024 Q5c OR, Summer 2024 Q5c

(c)
7 marksSame type asked in 3 of 5 papers

Define tree. In which order does a pre-order, in-order and post-order traversal visit the vertices of the ordered rooted tree shown in figure?

GTU Discrete Mathematics (3140708) Winter 2023 Question 5 OR (c) Diagram

Also asked in Summer 2026 Q5c OR, Winter 2024 Q5c

More Discrete Mathematics papers

Questions repeat across sessions. Working through two or three papers shows you which topics GTU asks most.

Summer 202670 marks · 5 questions · Solved
Winter 202570 marks · 5 questions
Winter 202470 marks · 5 questions
Summer 202470 marks · 5 questions

Other Semester 4 subjects

BE Computer

2140706
Numerical and Statistical Methods for Computer Engineering
2140707
Computer Organization
3140702
Operating System
3140705
Object Oriented Programming -I

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About this paper

Official Gujarat Technological University (GTU) examination paper for Discrete Mathematics (DM) (Winter 2023, B.E. · Computer Engineering, Sem 4). Features complete 70-mark regular & remedial examination pattern, official marking distribution across all 5 questions, and direct 1-click official PDF download.

Transcribed for student exam preparation from Gujarat Technological University official examination archives. Questions, syllabus guidelines, and curriculum marking schemes remain the intellectual property of Gujarat Technological University.

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