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MCS-013 Solved Assignment January 2026 | Discrete Mathematics | IGNOU BCA

QQ1. Q1.

(a)) What is Set? Explain use of Sets in problem solving. (150 words)
(b)) What is a proper subset ? Write the number of proper subsets of the Set {2, 3, 4, 5, 6, 7, 8, 9} (100 words)
(c)) Make the truth table for the following. i) p→~(r^q)^(~pvr) ii) p→ (r∨ q) ∨ (~p^~r) iii) p→ (r∨ ~q) (150 words)
(d)) Give geometric representation for the following. i) {5, -3) x (-1, 2) ii) {-1, -3) x (-2, -4) (100 words)

Key Points:

A Set is a well-defined collection of distinct objects (elements).
Sets are used for data organization, logic, and problem structuring.
A proper subset excludes the original set itself; formula is 2ⁿ - 1.
The set {2,3,4,5,6,7,8,9} has 8 elements, yielding 255 proper subsets.

Answer:

QQ2. Q2.

(a)) Draw Venn diagram to represent the following. i) (A∩BCC) ii) (A∩B∩C) iii) (A∪B∪C) (150 words)
(b)) Write down a suitable mathematical statement that the following symbolic properties can represent. i) (∀x) (∀y) (∃z)P ii) ∀(x) ∀ (y) Q (100 words)
(c)) Show whether √2 is rational or irrational. (100 words)
(d)) Explain proof by contradiction with the help of an example. (150 words)

Key Points:

Venn Diagram (A∩BCC): Represents elements in A and B, but not in C.
Venn Diagram (A∩B∩C): Represents elements common to all three sets A, B, and C.
Venn Diagram (A∪B∪C): Represents all elements present in A, B, or C, or any combination.
Universal Quantifier (∀): 'For all' or 'For every' elements in a domain.

Answer:

QQ3. Q3.

(a)) Explain use of the inclusion-exclusion principle with an example. (100 words)
(b)) Construct logic circuites for the following Boolean expressions: i) (x+yz) + (yz)' + (z'y) ii) (x'y) (xz') (y'z) + xyz (150 words)
(c)) What is a tautology? If P and Q are statements, show whether the statement (P → Q) v (~ P) is a tautology or not. (150 words)
(d)) Explain the symmetric difference of sets with the help of real-life examples. (100 words)

Key Points:

Inclusion-Exclusion Principle: |A ∪ B| = |A| + |B| - |A ∩ B| for two sets.
Logic circuits use AND, OR, NOT gates to visually represent Boolean expressions.
A tautology is a compound proposition that is always true in its truth table.
Truth tables verify tautologies by evaluating all possible truth assignments for components.

Answer: This response comprehensively addresses the various aspects of discrete mathematics as outlined in the question, covering fundamental concepts such as the inclusion-exclusion principle, logic circuit construction, tautologies, and symmetric differences of sets. Each section provides definitions, relevant formulas or methods, and illustrative examples or step-by-step solutions to ensure a thorough understanding of the topics. The answers adhere to specified word limits and formatting guidelines, ...

QQ4. Q4.

(a)) How many words can be formed using the letters of “EXCEPTIONAL”, using each letter at most once? i) If each letter must be used, ii) If some or all the letters may be omitted. (100 words)
(b)) What is a relation? Explain with an example. What are the different types of relations? Explain with an example for each. (400 words)

Key Points:

A relation R from set A to B is a subset of the Cartesian product A × B.
Reflexive relation: Every element relates to itself (a,a) ∈ R.
Symmetric relation: If (a,b) ∈ R, then (b,a) ∈ R.
Transitive relation: If (a,b) ∈ R and (b,c) ∈ R, then (a,c) ∈ R.

Answer: This question assesses fundamental concepts in combinatorics (permutations) and set theory (relations), which are core components of Discrete Mathematics (MCS-013). Part (a) involves calculating the number of possible word formations under different conditions, while Part (b) delves into defining relations and exploring their various types with suitable examples.

QQ5. Q5.

(a)) How many different professional committees of 8 people can be formed, each containing at least 4 Managers, at least 2 Public Servants and 2 IT Professionals from a list of 8 Managers, 6 Public Servants and 8 IT Professionals? (150 words)
(b)) A and B are mutually exclusive events such that P(A) = 1/3 and P(B) = 1/4 . Find P (A∪B). What is the probability of P(A∩B), and why? (100 words)
(c)) What is Pascal's triangle? Explain. (100 words)
(d)) Explain how to find the inverse of a function with the help of an example. (150 words)

Key Points:

Committee formation involves multiplying combinations for fixed roles and summing valid distributions for 'at least' conditions.
Mutually exclusive events imply P(A∩B) = 0, and their union probability is P(A∪B) = P(A) + P(B).
Pascal's triangle generates binomial coefficients, where each number is the sum of the two above it.
Rows of Pascal's triangle give coefficients for the expansion of (x+y)ⁿ.

Answer:

QQ6. Q6.

(a)) How many ways are there to distribute 25 district items into 7 distinct boxes with: i) At least three empty boxes. ii) No empty box. (100 words)
(b)) Explain properties of Set. (100 words)
(c)) Three Sets A, B and C are: A = {1,7, 8, 9, 13, 15, 17}, B = { 1,2, 3, 4, 5, 6, 8, 9, 10 } and C {1, 2, 3, 5, 7, 9, 10, 11, 13}. Find A∪B∩C; A ∩~B~C; A∩B∪C and (A~~C). (200 words)
(d)) Explain circular permutation with an example. (100 words)

Key Points:

Distributing 'n' distinct items into 'm' distinct boxes with no empty box uses PIE: Σ_{i=0 to m} (-1)^i * C(m, i) * (m-i)^n.
For 'at least k empty boxes' in distinct item/box distribution, sum C(m, j) * [surjective map to m-j boxes] for j=k to m-1.
Set properties like Idempotent (A∪A=A), Commutative (A∪B=B∪A), Associative ((A∪B)∪C=A∪(B∪C)), and Distributive (A∩(B∪C)=(A∩B)∪(A∩C)) are fundamental.
Set operations (union, intersection, complement) follow specific precedence (intersection before union) and require a universal set for complements.

Answer: Discrete Mathematics is a foundational subject in computer science, dealing with discrete elements. This response addresses key combinatorial problems, fundamental set theory properties, practical set operations, and circular permutations, all central topics within the MCS-013 curriculum. The distribution of distinct items into distinct boxes involves combinatorial counting principles, often employing the Principle of Inclusion-Exclusion (PIE) or Stirling numbers of the second kind for scenario...

QQ7. Q7.

(a)) Compare predicate and proposition logic. (100 words)
(b)) What is inductive logic? How is it used in problem-solving? Explain with an example. (150 words)
(c)) What is a function? Explain different types of functions with examples. (150 words)
(d)) Write the following statements in symbolic form: i) Mr. Ravi is thin but healthy. ii) Either do not eat unhealthy food or be ready to visit the doctor. (100 words)

Key Points:

Propositional logic uses variables for whole statements; predicate logic analyzes internal structure with predicates and quantifiers.
Inductive logic generalizes from specific observations to probable conclusions, useful for hypothesis generation and pattern recognition.
A function maps each domain element to exactly one codomain element (f: A → B).
One-to-one (injective) functions map distinct domain elements to distinct codomain elements.

Answer:

QQ8. Q8.

(a)) Find the inverse of the following functions f(x) = (x²+2)/(x-3) x=3 (150 words)
(b)) What is a Boolean function? Explain with an example. (100 words)
(c)) Show for an integer greater than zero 2ⁿ>=n+1 (100 words)
(d)) Write the inverse and contrapositive for these sentences: i) If it does not rain, then you will go to market. ii) If you are not honest, you are harmful to society. (150 words)

Key Points:

Inverse functions require the original function to be one-to-one (injective).
Algebraic solution for inverse of f(x)=(x²+2)/(x-3) involves quadratic formula.
Boolean function maps binary inputs to a single binary output (e.g., AND gate).
Mathematical Induction proves statements for all positive integers (Base, Hypothesis, Step).

Answer: This response addresses key concepts in Discrete Mathematics, covering inverse functions, Boolean functions, mathematical induction, and logical equivalences. Understanding these topics is crucial for building a strong foundation in computational and mathematical reasoning, which are central to the MCS-013 course. The calculation for the inverse function demonstrates algebraic manipulation and highlights conditions for invertibility. The explanation of Boolean functions introduces fundamental b...

MCS-013 Solved Assignment — January 2026 / July 2025

Discrete Mathematics | IGNOU Bachelor of Computer Applications (BCA)

MCS-013—January 2026 / July 2025

Discrete Mathematics

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MCS-013 Assignment Questions — January 2026

QQ1. Q1.

QQ2. Q2.

QQ3. Q3.

QQ4. Q4.

QQ5. Q5.

QQ6. Q6.

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QQ8. Q8.