Mumbai University Solved Question Papers
Discrete Structures and Graph Theory
Previous Year Question Paper with Solution
Engineering · Sem 3
Dec 2025 Examination
munotes.in
Mumbai
Engineering Sem 3 Discrete Structures and Graph Theory Dec 2025 Question Paper with Solutions
Get access to whole semester resourcesSemester Pass
First published on munotes.in on 5 November 2026.
Published by munotes.in, Mumbai.
Model answers written and edited by the munotes.in editorial desk.
Passages from this volume may be quoted, in print, online or by an AI system, with credit: name munotes.in and link to this volume's page. The volume may not be reproduced as a whole. Full terms at munotes.in/content-license.
munotes.in is an independent study resource for students of the University of Mumbai. It is not affiliated with the University of Mumbai, and is not endorsed by it.
The University does not publish an official answer key for this paper. The answers in this volume are model answers, written to show how a full-mark answer is built. They are a study aid, not an authority on what an examiner marked.
The question paper reproduced here is the paper as set by the University of Mumbai at the Dec 2025 examination.
Contents
The questions below are the paper as the University of Mumbai set it at the Dec 2025 examination, in the order it was set.
MarksPage
MarksPage
MarksPage
The Paper as Set
The questions in this volume are the questions asked at the Dec 2025 examination, reproduced as the University of Mumbai set them, in the order it set them. Nothing has been reworded, added or left out. Only the answers are ours. See the original question paper.
Duration 3 hours · Total marks 60 · 15 questions answered
Instructions printed on the paper
- N.B. (1) Q.1 is compulsory. (2) Attempt any three questions from the remaining. (3) Figures to the right indicate full marks. (4) Assume suitable data wherever necessary.
How to use this volume
Solve the paper first, under exam conditions and against the clock. Then read the answers here and mark your own. Reading a solution before attempting the question feels productive and teaches very little, because recognising an answer is not the same as being able to write one.
Q.1
Q.1
Attempt any three out of six sub questions · 15 Marks
a)A student-feedback system groups teacher responses using the rule: Two responses x and y are "similar" if they fall in the same satisfaction band (e.g., 1-2 = Low, 3 = Medium, 4-5 = High). (i) Using definitions of reflexive, symmetric, transitive, determine whether "similarity" is an equivalence relation.[5]
(ii) Explain the classroom-level meaning of each property. (iii) Discuss how violating any property affects AI-based feedback analytics and dashboard reports.
Answer
For full marks, cover: the relation and its bands, the test of each of the three properties, the verdict and the equivalence classes, what each property means in a classroom, and what breaks in the analytics when a property fails.
The relation
Let A = {1, 2, 3, 4, 5} be the possible ratings. The bands are Low = {1, 2}, Medium = {3} and High = {4, 5}. Write b(x) for the band of a rating x. Two ratings are similar, x ~ y, exactly when b(x) = b(y).
Testing the three properties
- Reflexive. For every x in A, b(x) = b(x). So x ~ x for all five ratings, and the relation is reflexive.
- Symmetric. If x ~ y, then b(x) = b(y). So b(y) = b(x), which means y ~ x. The relation is symmetric.
- Transitive. If x ~ y and y ~ z, then b(x) = b(y) and b(y) = b(z). Hence b(x) = b(z), so x ~ z. The relation is transitive.
All three properties hold, so "similarity" is an equivalence relation on A. As a set of ordered pairs it has 4 + 1 + 4 = 9 pairs. Its equivalence classes are the bands themselves: [1] = {1, 2}, [3] = {3} and [4] = {4, 5}. The three classes are disjoint and together cover A, so they form a partition of the ratings.
What each property means in a classroom
| Property | Classroom meaning |
|---|---|
| Reflexive | Every response is similar to itself, so no response is left without a group. |
| Property | Classroom meaning |
|---|---|
| Symmetric | If Riya's response matches Aman's, then Aman's matches Riya's. The order of comparison does not matter. |
| Transitive | If two responses each match a third, they match each other. A band never splits into overlapping groups. |
When a property is violated
- If reflexivity failed, some responses would belong to no group. The dashboard totals would then fall short of the number of responses collected.
- If symmetry failed, the grouping would depend on which response is compared first. Two runs of the same analysis could report different clusters.
- Transitivity fails for a rule such as "ratings differ by at most 1". Then 2 ~ 3 and 3 ~ 4, but 2 is not similar to 4. The groups overlap, one response is counted in two bands, and the percentages add up to more than 100.
- An AI model trained on overlapping labels learns noise. Its sentiment summaries and trend charts then mislead teachers and heads of department.
Conclusion. Grouping by band is reflexive, symmetric and transitive, so it is an equivalence relation. It splits the five ratings into three classes, Low, Medium and High. That partition is what lets every dashboard count each response exactly once.
The rest of the answers
The first answer is free. The rest come with the B.E. (Computer Engineering) Semester 3 solved papers.
You have read the question paper and its first answer in full. Buy the solved papers once and you can read every answer of every solved paper in this semester.
Solved papers: ₹499 Already bought it? Sign in
The question paper itself stays free, and so does the syllabus.
Found an error in this volume? Report it and we will check it against the paper.