Gujarat Technological UniversitySummer 2024 examination

GTU 3140708 Discrete Mathematics (DM) Summer 2024 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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9 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 marks

Define injective function. Given 𝐴 = {2, 5, 6}, 𝐵 = {3, 4, 2}, find (𝐴 − 𝐵) and (𝐵 − 𝐴).

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

Determine the relation ≤ (less than or equal) on the set ℤ of integers are reflexive, symmetric, anti-symmetric, transitive.

Also asked in Winter 2024 Q1b, Winter 2023 Q1b

(c)
7 marks
(
i)Check whether the function 𝑓(𝑥) = 𝑥³ − 2, for 𝑥 ∈ ℝ is invertible function. If so, find 𝑓−1(𝑥).
ii)Prove that a tree with 𝑛 vertices has 𝑛 − 1 edges.3 Marks

Q.2

14 Marks
(a)
3 marksSame type asked in 4 of 5 papers
Identify the statement (¬𝑞⋀(𝑝 → 𝑞)) → ¬𝑝 is tautology or contradiction without
constructing the truth table.

Also asked in Winter 2025 Q3a, Winter 2025 Q3b, Winter 2024 Q2b, Winter 2023 Q2a

(b)
4 marks

Let 𝐺 be the subset of 2 × 2 real matrices with a nonzero determinant. Check whether 𝐺 is group under matrix multiplication. If so, is it abelian group?

(c)
7 marksSame type asked in 3 of 5 papers
(
i)Symbolize the expression “John is a bachelor and this painting is red”.
ii)Express the following using predicate, quantifier and logical connectives. Also verify the validity of the consequence. Everyone who graduates gets a job. Ram is graduated. Therefore, Ram got a job.3 Marks

Also asked in Winter 2024 Q2c OR, Winter 2023 Q2c OR

(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
𝑝 → 𝑟
∴ 𝑝 → 𝑞 ∨ 𝑟

Also asked in Winter 2024 Q2c, Winter 2023 Q2c

Q.3

14 Marks
(a)
3 marks

Let 𝑔 be a homomorphism from a group (𝐺,∗) to a group (𝐻, ∆). Show that 𝑔(𝑒𝐺) = 𝑒𝐻 and for any 𝑎 ∈ 𝐺, 𝑔(𝑎⁻¹) = (𝑔(𝑎))⁻¹.

(b)
4 marks
Prove that : 𝐴 ∪ (𝐵 ∩ 𝐶) = (𝐴 ∪ 𝐵) ∩ (𝐴 ∪ 𝐶).
(c)
7 marks
(
i)Prove that every cyclic group is abelian.
ii)Consider the set of positive integers ℕ. Check which of (ℕ, +) and (ℕ,×) are semigroup and which are monoid?3 Marks
(a)
3 marks
Find left cosets and right cosets of 𝐻 = {0,3} in the group (ℤ6, +6).
(b)
4 marks
(
i)Suppose repetitions are not allowed, how many four digit numbers can be formed from six digits 1,2,3,5,7,8?
ii)How many of such numbers less than 4000?
iii)How many in
i)are even?
iv)How many in
i)are divisible by 10?
(c)
7 marksSame type asked in 2 of 5 papers

Show that (𝑅, +, ×) is an integral domain, where 𝑅 = {𝑎 + 𝑏√5 / 𝑎, 𝑏 ∈ ℤ }.

Also asked in Winter 2025 Q5c OR

Q.4

14 Marks
(a)
3 marks

Let 𝑆 = {1, 2, 3, 4} and 𝑅 = {(1,1), (1,4), (2,2), (2,3), (3,2), (3,3), (4,1), (4,4)} . Draw the graph of 𝑅 and hence write partition of 𝑆.

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

Define Lattice. Draw the Hasse diagram of (𝑆12, 𝐷), where 𝐷 is the relation of “division” in ℕ such that for any 𝑎, 𝑏 ∈ ℕ , 𝑎𝐷𝑏 iff 𝑎 divides 𝑏 and 𝑆12 is the set of all divisors of 12.

Also asked in Summer 2026 Q2c, Winter 2025 Q3c, Winter 2025 Q4a OR, Winter 2024 Q4c OR

(c)
7 marks
Let ⟨𝐿, ≤⟩ be a lattice. Show that for 𝑎, 𝑏, 𝑐 ∈ 𝐿, following inequalities holds.
i)𝑎 ⊕ (𝑏 ∗ 𝑐) ≤ (𝑎 ⊕ 𝑏) ∗ (𝑎 ⊕ 𝑐) and
ii)𝑎 ∗ (𝑏 ⊕ 𝑐) ≥ (𝑎 ∗ 𝑏) ⊕ (𝑎 ∗ 𝑐).
(a)
3 marks
Let the POSET (ρ(A), ≤) where A = {a, b, c}, relation is subset. Find
i)Upper bound of {{ }, {a}, {c}},
ii)GLB of {{ }, {a}, {c}}, if exist,
ii)LUB of {{ }, {a}, {c}}, if exist.
(b)
4 marksSame type asked in 4 of 5 papers
Solve 𝑎𝑛 = 𝑎𝑛−1 + 𝑎𝑛−2, 𝑎0 = 0, 𝑎1 = 1.

Also asked in Winter 2025 Q3c OR, Winter 2024 Q4b OR, Winter 2023 Q4c OR

(c)
7 marks

Given the relation matrices 𝑀𝑅 and 𝑀𝑆, find 𝑀𝑅∘𝑆, 𝑀𝑅̃ , 𝑀𝑆̃ , 𝑀𝑅∘𝑆 ̃ , and show that 𝑀𝑅∘𝑆 ̃ = 𝑀𝑆̃ ∘ 𝑀𝑅̃ . MR = [ 1 0 1 1 1 0 1 1 1 ] and MS = [ 1 0 0 1 0 1 0 1 0 1 0 0 1 1 0 ].

Q.5

14 Marks
(a)
3 marks
Define Isolated node, Binary tree and Regular graph.
(b)
4 marksSame type asked in 3 of 5 papers
Check whether the following graphs are isomorphic or not.

Also asked in Summer 2026 Q3c, Winter 2024 Q5b

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

Define path matrix. Warshall's algorithm to obtain path matrix from the adjacency matrix of following graph

Also asked in Summer 2026 Q3c OR, Winter 2024 Q5c OR, Winter 2023 Q5b OR

(a)
3 marks

A graph 𝐺 has 15 edges, 3 vertices of degree 4 and other vertices of degree 3. Find the number of vertices in 𝐺.

(b)
4 marks
Find all the node base of the given digraph. Also find 𝑑(𝑉3, 𝑉6), 𝑑(𝑉6, 𝑉3).
(c)
7 marks

Draw binary trees whose post-order produced the string d-e-c-g-j-h-f-b-l-n-q-r-p-m-k-a and pre-order produced the string a-b-d-h-e-i-j-c- f-g-k and in-order produced the string h-d-b-i-e-j-a-f-c-k-g.

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
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About this paper

Official Gujarat Technological University (GTU) examination paper for Discrete Mathematics (DM) (Summer 2024, 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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