BCA 207T Discrete Mathematics Notes cover all 4 units of this GGSIPU BCA Semester 3 paper in 81 pages: 129 solved examples, 48 two-mark questions with answers, 32 ten-mark exam-style questions and 25 labelled diagrams. They follow the 2024-25 BCA scheme, which applies to the batch admitted in 2024-25. Order the printed book on WhatsApp.
Written by Prateek Talwar, Founder, Unnati Educations · Reviewed by Ritik Rawat, B.Tech, Senior Full Stack Developer · Checked against BCA Programme Common Scheme and Syllabus (w.e.f. A.S. 2024-2025), ipu.ac.in · Updated 12 October 2026 · Editorial policy
Scheme: BCA 2024-25 scheme (paper code BCA 207T) | Applies to: batch admitted in academic session 2024-25 | Verified against: BCA Programme Common Scheme and Syllabus (w.e.f. A.S. 2024-2025), second-year syllabus approved 01.08.2025
On this page
- BCA 207T at a glance: credits, units and exam pattern
- BCA 207T notes unit-wise
- BCA 207T exam corner
- Excerpt from page 75: Euler circuits
- Do the notes cover the full syllabus?
- Who writes and checks the notes?
- BCA 207T and the Sem 3 papers it connects to
- How to write BCA 207T answers
- FAQs
- Get the full notes on WhatsApp
BCA 207T Discrete Mathematics at a glance: GGSIPU BCA Sem 3 credits, units and exam pattern
BCA 207T is the only core theory paper of the 2024-25 BCA Semester 3 without a paired lab. It sits with the other GGSIPU BCA Semester 3 papers of that scheme.
| Code and title | BCA 207T · Discrete Mathematics |
|---|---|
| Course and semester | BCA, Semester 3 |
| Category and credits | Core Course Theory (CCT) · 4 credits · L-T-C 4-0-4 (4 lecture hours a week) |
| Marks | 100: 40 internal + 60 end-term |
| Applies to | Batch admitted in academic session 2024-25 |
| End-term pattern | Q1 compulsory: 10 × 2 marks, at least 2 per unit; then one 10-mark question per unit, from two choices (20 + 40 = 60 marks) |
| Units and teaching hours | 4 units: 13 + 10 + 11 + 10 = 44 hours |
| Paired lab | None |
Batch check: from the 2025-26 batch, the new BCA scheme uses BCA 207T for Introduction to Data Analytics and moves Discrete Mathematics to Semester 1 as BCA 101. So a 2025-26 student looking for GGSIPU BCA Sem 3 Discrete Mathematics notes will not find this paper in Semester 3. See the BCA and BCA-DS common scheme (w.e.f. 2025-26) before you buy.
BCA 207T Discrete Mathematics Notes unit-wise: 129 solved examples across 4 units
Each unit of the 81-page book ends with a Quick Revision Sheet and an Exam Corner on the 2 + 10 mark pattern. Notes for every other BCA paper sit under GGSIPU BCA notes.
The official PDF prints topic heads but no unit titles, so the headings below use the notes' unit titles.
BCA 207T Unit 1: Sets, Relations and Functions
Pages 4 to 24, 14 sections: set forms, power sets and the laws of sets; relation properties, equivalence classes and composition by Boolean matrix product; one-one, onto and inverse functions. 34 solved examples, 6 figures, 12 two-mark answers, 8 ten-mark questions.
| Kind of function A → B | How many |
|---|---|
| All functions | nᵐ |
| One-one functions (m ≤ n) | ⁿPₘ = n!/(n − m)! |
| Bijections (m = n) | n! |
| Onto functions from a 3-set to a 2-set | 2³ − 2 = 6 |
Quick look: counting functions when n(A) = m and n(B) = n (from page 19)
- Kind of function A → B
- All functions
- How many
- nᵐ
- Kind of function A → B
- One-one functions (m ≤ n)
- How many
- ⁿPₘ = n!/(n − m)!
- Kind of function A → B
- Bijections (m = n)
- How many
- n!
- Kind of function A → B
- Onto functions from a 3-set to a 2-set
- How many
- 2³ − 2 = 6
BCA 207T Unit 2: Propositional Logic and Methods of Proof
Pages 25 to 43, 10 sections: truth tables, tautology versus contingency, laws of logic, ∀ and ∃ with their negation, then direct, contrapositive, contradiction, case-wise and induction proofs. The largest unit by worked problems: 35 solved examples, 12 two-mark answers, 8 ten-mark questions.
| Rule | Premises | Conclusion |
|---|---|---|
| Modus Ponens | p, p → q | q |
| Modus Tollens | ¬q, p → q | ¬p |
| Hypothetical Syllogism | p → q, q → r | p → r |
| Disjunctive Syllogism | p ∨ q, ¬p | q |
Quick look: rules of inference with premises and conclusion (from page 33)
- Rule
- Modus Ponens
- Premises
- p, p → q
- Conclusion
- q
- Rule
- Modus Tollens
- Premises
- ¬q, p → q
- Conclusion
- ¬p
- Rule
- Hypothetical Syllogism
- Premises
- p → q, q → r
- Conclusion
- p → r
- Rule
- Disjunctive Syllogism
- Premises
- p ∨ q, ¬p
- Conclusion
- q
BCA 207T Unit 3: Partial Order Relations and Lattices
Pages 44 to 62, 13 sections: POSETs and chains, maximal and minimal elements, lub and glb, lattices as order and as algebra, then bounded, distributive and complemented lattices with the M₃ and N₅ test. 33 solved examples, 6 figures, 12 two-mark answers, 8 ten-mark questions.
| Lattice | Atoms |
|---|---|
| (D₃₀, |) | 2, 3, 5 (the prime divisors of 30) |
| (D₁₂, |) | 2, 3 |
| (P({a, b, c}), ⊆) | {a}, {b}, {c} |
| Chain (D₈, |) = {1, 2, 4, 8} | 2 only |
Quick look: atoms of four standard lattices (from page 55)
- Lattice
- (D₃₀, |)
- Atoms
- 2, 3, 5 (the prime divisors of 30)
- Lattice
- (D₁₂, |)
- Atoms
- 2, 3
- Lattice
- (P({a, b, c}), ⊆)
- Atoms
- {a}, {b}, {c}
- Lattice
- Chain (D₈, |) = {1, 2, 4, 8}
- Atoms
- 2 only
BCA 207T Unit 4: Graphs
Pages 63 to 81, 11 sections: degree and the Handshaking Theorem, graph types, subgraphs, isomorphism invariants, complements, adjacency and incidence matrices, walks and paths, then Hamiltonian graphs with Dirac's theorem. 27 solved examples, 13 figures, 12 two-mark answers, 8 ten-mark questions.
| Graph | Formula | Edges |
|---|---|---|
| K₇ | n(n − 1)/2 = 7 × 6/2 | 21 |
| K₃,₄ | m × n = 3 × 4 | 12 |
| 3-regular graph on 8 vertices | nk/2 = 8 × 3/2 | 12 |
| 3-regular graph on 7 vertices | 7 × 3/2 = 10.5 | Not possible |
Quick look: edge counts from the standard formulas (from page 68)
- Graph
- K₇
- Formula
- n(n − 1)/2 = 7 × 6/2
- Edges
- 21
- Graph
- K₃,₄
- Formula
- m × n = 3 × 4
- Edges
- 12
- Graph
- 3-regular graph on 8 vertices
- Formula
- nk/2 = 8 × 3/2
- Edges
- 12
- Graph
- 3-regular graph on 7 vertices
- Formula
- 7 × 3/2 = 10.5
- Edges
- Not possible
BCA 207T exam corner: 48 two-mark answers and 32 ten-mark questions by unit
Each Exam Corner has twelve "2 Marks" questions with brief answers and eight "10 Marks" questions with section pointers.
| Unit | 2 Marks (with answers) | 10 Marks (exam-style) | Solved examples |
|---|---|---|---|
| I · Sets, Relations and Functions | 12 | 8 | 34 |
| II · Propositional Logic and Methods of Proof | 12 | 8 | 35 |
| III · Partial Order Relations and Lattices | 12 | 8 | 33 |
| IV · Graphs | 12 | 8 | 27 |
| Total | 48 | 32 | 129 |
BCA 207T Exam Corner questions and solved examples per unit
- Unit
- I · Sets, Relations and Functions
- 2 Marks (with answers)
- 12
- 10 Marks (exam-style)
- 8
- Solved examples
- 34
- Unit
- II · Propositional Logic and Methods of Proof
- 2 Marks (with answers)
- 12
- 10 Marks (exam-style)
- 8
- Solved examples
- 35
- Unit
- III · Partial Order Relations and Lattices
- 2 Marks (with answers)
- 12
- 10 Marks (exam-style)
- 8
- Solved examples
- 33
- Unit
- IV · Graphs
- 2 Marks (with answers)
- 12
- 10 Marks (exam-style)
- 8
- Solved examples
- 27
- Unit
- Total
- 2 Marks (with answers)
- 48
- 10 Marks (exam-style)
- 32
- Solved examples
- 129
Four exam-style questions from the notes, one per unit:
- Unit I (2 Marks): How many relations are possible on a set with 3 elements? Pointer: A × A has 9 pairs, so 2⁹ = 512.
- Unit II (10 Marks): Test the validity of: if it rains, Ram skips college; if he skips college, he plays games; he does not play games; so it does not rain. Pointer: hypothetical syllogism, then Modus Tollens; valid.
- Unit III (2 Marks): Is (D₁₂, |) complemented? Pointer: no; 2 has no complement.
- Unit IV (2 Marks): Can a simple graph have degree sequence 4, 3, 3, 2, 1? Pointer: the degree sum 13 is odd, so no.
The BCA 207T notes answer 48 two-mark questions and set 32 ten-mark exam-style questions across 4 units, plus 129 solved examples.
More practice sits in the BCA 207T important questions.
Euler circuits and Euler's theorem in graph theory: excerpt from page 75 of the BCA 207T notes
Excerpt from page 75
Connected graphs
A graph is connected if there is a path between every pair of vertices. If not, it is disconnected, and its largest connected pieces are called components. In the Unit IV example where vertex v5 has no edge at all, v5 is cut off, so that graph has 2 components.
Euler trail and Euler circuit
An Euler trail (also called an Euler path) is a trail that uses every edge exactly once. An Euler circuit is a closed Euler trail: it starts and ends at the same vertex. A graph that has an Euler circuit is called an Euler (Eulerian) graph. A graph that has an Euler trail but no Euler circuit is called semi-Eulerian.
The easy way to picture it: you must draw the whole figure without lifting the pen and without going over any line twice.
Euler's theorem
- A connected graph has an Euler circuit if and only if every vertex has even degree.
- A connected graph has an Euler trail but no circuit if and only if exactly two vertices have odd degree. The trail must start at one odd vertex and end at the other.
- If more than two vertices have odd degree, there is no Euler trail at all.
Solved Example 21: why must every vertex have even degree in an Euler circuit?
Follow the Euler circuit around the graph. Every time it passes through a vertex v, it uses one edge to come in and another edge to go out, so each visit uses 2 edges at v.
The start vertex is paired as well: the first edge going out matches the last edge coming back. Since every edge is used exactly once, all the edges at v are used up in these in-and-out pairs.
So deg(v) is even for every vertex v. Hence proved. The converse also holds for connected graphs: if every degree is even, an Euler circuit exists.
Do the BCA 207T notes cover the full GGSIPU syllabus?
Yes. Every BCA 207T topic in the 2024-25 syllabus has its own section in the notes.
| Official unit (hours) | Official topics | Where in the notes | Status |
|---|---|---|---|
| Unit I (13) | Sets, subsets, set operations, algebra of sets, Cartesian product, applications; properties of relations, equivalence and partial order relations, composition, digraph and matrix; onto, into, one-one, composite and inverse functions | 1.1 to 1.14, pages 5 to 21 | Covered |
| Unit II (10) | Propositions, first order logic, truth tables, tautologies, algebra of propositions, implication, equivalence, predicates, quantifiers; direct, indirect, exhaustive-case and induction proofs | 2.1 to 2.10, pages 26 to 40 | Covered |
| Unit III (11) | POSETs, totally ordered sets, Hasse diagrams, chains, maximal and minimal elements, glb, lub, lattices, duality, atoms, sublattices, bounded, distributive, complemented and isomorphic lattices | 3.1 to 3.13, pages 45 to 59 | Covered |
| Unit IV (10) | Directed and undirected graphs, degree, pendent vertex, Handshaking Theorem; graph types; subgraph, isomorphism, complement; adjacency and incidence matrices; walk to cycle, Euler graph, seven bridges, Hamiltonian graph | 4.1 to 4.11, pages 64 to 78 | Covered |
Official BCA 207T units mapped to sections and pages in the notes
- Official unit (hours)
- Unit I (13)
- Official topics
- Sets, subsets, set operations, algebra of sets, Cartesian product, applications; properties of relations, equivalence and partial order relations, composition, digraph and matrix; onto, into, one-one, composite and inverse functions
- Where in the notes
- 1.1 to 1.14, pages 5 to 21
- Status
- Covered
- Official unit (hours)
- Unit II (10)
- Official topics
- Propositions, first order logic, truth tables, tautologies, algebra of propositions, implication, equivalence, predicates, quantifiers; direct, indirect, exhaustive-case and induction proofs
- Where in the notes
- 2.1 to 2.10, pages 26 to 40
- Status
- Covered
- Official unit (hours)
- Unit III (11)
- Official topics
- POSETs, totally ordered sets, Hasse diagrams, chains, maximal and minimal elements, glb, lub, lattices, duality, atoms, sublattices, bounded, distributive, complemented and isomorphic lattices
- Where in the notes
- 3.1 to 3.13, pages 45 to 59
- Status
- Covered
- Official unit (hours)
- Unit IV (10)
- Official topics
- Directed and undirected graphs, degree, pendent vertex, Handshaking Theorem; graph types; subgraph, isomorphism, complement; adjacency and incidence matrices; walk to cycle, Euler graph, seven bridges, Hamiltonian graph
- Where in the notes
- 4.1 to 4.11, pages 64 to 78
- Status
- Covered
Official sources: BCA Programme Common Scheme and Syllabus (w.e.f. A.S. 2024-2025) and the BCA 4-year curriculum and scheme (2024-25), which prints the marks.
Every course in GGSIPU notes is mapped to its official PDF the same way.
Who writes and checks the BCA 207T Discrete Maths notes?
- Prateek Talwar, author of these notes. Founder, Unnati Educations. Associate Company Secretary (ACS), ICSI. B.Com, University of Delhi (School of Open Learning), 2012. Author profile
- Ritik Rawat, subject reviewer. Senior Full Stack Developer, Unnati Educations. B.Tech. Reviews the BCA and B.Tech notes. Reviewer profile
The 2026-27 edition is mapped to the 2024-25 BCA scheme: 81 pages, a page-numbered index, four Quick Revision Sheets and four Exam Corners. Last updated 12 October 2026. See our editorial policy.
Anyone wanting BCA 207T Discrete Mathematics IPU notes that match the paper-setter pattern will find it in the "Remember" box on page 3. The same team prepares GGSIPU important questions for every course.
BCA 207T and the GGSIPU BCA Sem 3 papers it connects to
BCA 207T has no practical paper, so see the full GGSIPU BCA syllabus for how it fits the programme.
Semester 3 papers in the 2024-25 scheme: BCA 201T Python Programming, BCA 203T Dynamic Web Designing, BCA 205T Computer Organization and Architecture, labs 201P, 203P and 205P, BCA 261 Vocational Course, BCA 221T or 223T (GE-1) and BCA 291T Human Values and Ethics (NUES).
- Unit II laws of logic → Boolean algebra and logic gates in BCA 205T Unit I.
- Unit II connectives →
ifandwhileconditions in BCA 201T. - Unit I relations and Cartesian product → relational algebra in BCA 102T DBMS (Semester 2).
- Unit IV adjacency and incidence matrices → graph storage in BCA 106T DSA (Semester 2).
How to write BCA 207T answers: induction proofs, truth tables and Hasse diagrams
Two-thirds of the end-term marks sit in four 10-mark answers. The notes give a writing pattern for each type.
- Set identities: element method, LHS ⊆ RHS then RHS ⊆ LHS (page 9).
- Relations: give both matrix and digraph, then each property with a one-line reason (page 18).
- Truth tables: 2ⁿ rows, inner brackets first (page 28).
- Hasse diagrams: join each divisor upward only to itself times one prime (page 48).
- Graph terms: write your definition of path and cycle first (page 74).
For induction, page 39 gives a five-step layout: state P(n), check the basis, write the hypothesis, prove P(k + 1) while marking where the hypothesis is used, and conclude. Here it is on Solved Example 34 from page 40.
P(n): 7ⁿ − 1 is divisible by 6
Basis: 7¹ − 1 = 6, true
Hypothesis: 7ᵏ − 1 = 6m
Step: 7ᵏ⁺¹ − 1 = 7(6m + 1) − 1 = 6(7m + 1)
Hence by PMI, true for all n ∈ NPage 41 warns against skipping the basis step or assuming P(k + 1). Practise this format on GGSIPU BCA important questions.
BCA 207T Discrete Maths notes FAQs
Which batch does BCA 207T Discrete Mathematics apply to?
It applies to students admitted in academic session 2024-25, who follow the 2024-25 BCA scheme. In the newer scheme for batches from 2025-26, the code BCA 207T belongs to Introduction to Data Analytics, and Discrete Mathematics is taught in Semester 1 as BCA 101. Check the code and title on your own date sheet before you buy these notes.
What is the passing mark in BCA 207T?
BCA 207T is marked out of 100: 40 internal marks and 60 end-term marks, as printed in the BCA scheme of examination. The scheme does not print a separate pass percentage for core theory papers like this one. Minimum marks in the end-term exam, results and re-appear rules are set by GGSIPU Ordinance 11, so read it on ipu.ac.in or ask your college exam cell.
Which BCA 207T unit has the most solved examples?
Unit II, Propositional Logic and Methods of Proof, has the most: 35 solved examples, ranging from truth tables and laws of logic to argument validity and induction proofs. Unit I follows with 34, Unit III has 33 and Unit IV has 27. Every unit carries the same Exam Corner of 12 two-mark answers and 8 ten-mark questions.
Does BCA 207T have a lab or practical paper?
No. In the 2024-25 BCA scheme, BCA 207T is a 4-credit Core Course Theory paper with 4 lecture hours a week and no practical component. The Semester 3 labs are BCA 201P Python, BCA 203P Dynamic Web Designing and BCA 205P Computer Organization and Architecture. The notes are therefore written for the written exam: definitions, proofs, worked examples and diagrams.
How many diagrams do the BCA 207T notes have for Hasse diagrams and graphs?
The notes have 25 labelled figures: 6 in Unit I (Venn diagrams, arrow diagrams and a digraph), 6 in Unit III and 13 in Unit IV. Unit III also draws extra Hasse diagrams inside its solved examples for D₁₂, D₁₈ and P({a, b, c}). Unit IV figures include the Königsberg bridges model and the pentagon and pentagram drawn as one graph.
Get the full BCA 207T Discrete Mathematics Notes on WhatsApp
| Part | Pages |
|---|---|
| Cover and index | 1 to 3 |
| Unit I | 4 to 24 (21) |
| Unit II | 25 to 43 (19) |
| Unit III | 44 to 62 (19) |
| Unit IV | 63 to 81 (19) |
Pages per unit in the BCA 207T notes book
- Part
- Cover and index
- Pages
- 1 to 3
- Part
- Unit I
- Pages
- 4 to 24 (21)
- Part
- Unit II
- Pages
- 25 to 43 (19)
- Part
- Unit III
- Pages
- 44 to 62 (19)
- Part
- Unit IV
- Pages
- 63 to 81 (19)
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The printed 81-page book of Discrete Mathematics notes in English has 129 solved examples, 48 two-mark answers and 32 ten-mark questions. Delivery details are shared on WhatsApp.
Unnati Educations is an independent education provider based in Vikaspuri, Delhi. We are not affiliated with, authorised by, or endorsed by Guru Gobind Singh Indraprastha University (GGSIPU). 'GGSIPU' and 'IPU' are used only to describe the syllabus our notes follow. For official information, visit ipu.ac.in.
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