Last Minute Notes – Discrete Mathematics (original) (raw)

Last Updated : 26 Aug, 2025

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Discrete mathematics typically constituting 10% of the total marks. Some of the important topics of Discrete Mathematics are :

**Read : Last Minute Notes on all subjects - For GATE Exams

Propositional Logic

Propositional Logic deals with propositions (simple declarative statements) that can be true or false.

**Operators in Propositional logic: Words or symbols that modify or join statements to form more complex statements.

Operator Term Precedence
¬ Negation ( NOT ) 1
∧ Conjunction ( AND ) 2
∨ Disjunction ( OR ) 3
**→ Implication 4
↔ Double Implication 5

**Propositional Statements and their Meanings

Statement Meaning
if p, then q p implies q
if p, q p only if q
p is sufficient for q a sufficient condition for q is p
q if p q whenever p
q when p q is necessary for p
a necessary condition for p is q q follows from p
q unless ¬ p q provided that p

**Truth tables for propositional Statements.

p q ¬p p ∧ q p ∨ q p **→ q p ↔ q
T T F T T T T
T F F F T F F
F T T F T T F
F F T F F T T

Laws in Propositional Logic

**De Morgan’s Law

**Some other laws :

Conditional Statements in Propositional Logic

1. Implication : If p (the antecedent) is true, then q (the consequent) must also be true. If p is false, the statement is true regardless of q's value.

**Representation : p **→ q
**Example: If it rains (p), then the ground will be wet (q)

**2. Converse : It represents the reverse of a conditional statement. If q is true, then p must be true.

**Representation : For a statement **p → q its converse is represented as **q → p
**Example: If the ground is wet (q), then it must have rained (p).

**3. Contrapositive : Negates and reverses the original conditional statement. If q is false, then p is false.

**Representation : For a statement **p → q its contrapositive is represented as (¬q → ¬p)
**Example: If the ground is not wet (¬q), then it did not rain (¬p).

  1. Inverse : Negates both the antecedent and the consequent of the original conditional statement. If p is false, then q is false.

**Representation : For a statement **p → q its inverse is represented as (**¬p → ¬q)
**Example: If the ground is not wet (¬q), then it did not rain (¬p).

**5. Biconditional: In a biconditional p is true if and only if q is true. Both p and q must either be true or false together.

**Representation : For a statement **p → q its biconditional is represented as (**p ↔ q)
**Example: It rains if and only if the ground is wet.

**Types of propositions based on Truth values

**1. Tautology– A **tautology is a logical proposition or formula that is always **true, regardless of the truth values of its components.
**Example: p ∨¬p

**2. Contradiction– A **contradiction is a logical proposition or formula that is always **false, regardless of the truth values of its components.
**Example: p ∧¬p

**3. Contingency – A **contingency is a logical proposition or formula that is **neither always true nor always false; its truth value depends on the truth values of its components.
**Example: p ∧q

**Rules of Inference

A **rule of inference is a logical rule that specifies valid ways to derive conclusions from premises in a formal argument.

Predicate Logic

**Predicates are expressions in logic that represent properties, conditions, or relationships involving one or more variables.
**Example : P(x) ∧ Q(x)

**Quantifiers in predicate : Quantifiers like ∀(for all) and ∃(there exists) are used with predicates to create logical statements:

**Important equivalences involving quantifiers

**Predicate logic follows all the rules of **propositional logic, but it extends those rules to include variables, quantifiers, and predicates.

Combinatorics

**Combinatoricsis the branch of mathematics that studies the counting, arrangement, and combination of objects within a set under specific rules.

**Permutation: A permutation of a set of distinct objects is an ordered arrangement of these objects.

\begin{flalign*} P(n, r) &= n * (n-1) * ... * (n-r+1)\\ &= \frac{n * (n-1) * ... * (n-r+1) * (n-r) *...* 2 * 1}{(n-r) * (n-r-1) * .... * 2 * 1}\\ &= \frac{n!}{(n-r)!} \end{flalign*}

**Combination: A combination of a set of distinct objects is just a count of the number of ways a specific number of elements can be selected from a set of a certain size. The order of elements does not matter in a combination. and it is given by

\begin{flalign*} C(n, r) &= \frac{P(n, r)}{P(r, r)}\\ &= \frac{n!}{(n-r)!} * \frac{1}{r!}\\ &= \frac{n!}{r!(n-r)!}& \end{flalign*}

**Binomial Coefficients: The combination of r items form a set of n elements is denoted by nCr. This number is also called a binomial coefficient since it occurs as a coefficient in the expansion of powers of binomial expressions.
Let x and y be variables and n be a non-negative integer. Then

\begin{flalign*} (x+y)^n &= \sum_{j=0}^{n} \binom{n}{j} x^{n-j}y^j\\ &= \binom{n}{0}x^{n} + \binom{n}{1}x^{n-1}y +...+ \binom{n}{n-1}xy^{n-1} + \binom{n}{n}y^{n} \end{flalign*}

**The binomial expansion using Combinatorial symbols

(a+b)^n = ^nC_0 a^n b^0 + ^nC_1 a^{n-1} b^1 + ^nC_2 a^{n-2} b^2 .. + ^nC_{n-k} a^k b^{n-k} .. +^nC_n a^0 b^n

**Number of elements in the Union of Finite sets

For a number of finite sets, A1, A2, . . . , An, the number of elements in the union A1 U A2 U ... U An is given by :

****| A** 1 **U A 2 **U ... U A n | = ∑ i | A i ****| - ∑** i<j | A i **⋂ A j | + ∑ i<j<k | A i **⋂ A j ⋂ A k | - .... + (-1) n+1 | A 1 **⋂ A 2 ⋂ A 3 ⋂ .... ⋂A n |

**Important Binomial Expansion Formulas

Some of the important Binomial Expansion formulas are:

Set Theory

**Set is an unordered collection of objects, known as elements or members of the set. Example { 1,3,5,4,7, 9,2 }
An element ‘a’ belong to a set A can be written as ‘a ∈ A’, ‘a ∉ A’ denotes that a is not an element of the set A.

asubsetB

**Set Operations

**Laws Related to Set Operations

**Multiset : A **multiset is a collection of elements where duplicates are allowed, and the number of times an element appears (its "multiplicity") matters. Unlike a regular set, a multiset can have repeated elements. **Example : {a, a, b, c, c, c} is a multiset where multiplicity of c is 3.
**Note : Size of multiset is equal to the sum of all the multiplicities.

**Relations

A **relation R from a set A to a set B is a subset of the Cartesian product A×B. That is: R ⊆ A×B . Each element of R is an ordered pair (a, b), where a ∈ A and b ∈ B, and a is related to b under R.

Number of possible relations on a set A = 2^{n^2} where n is the number of element of set A.
For a relation (x, y) ∈ R
**Domain of R = all possible values of x
**Range of R = all possible values of y

**Inverse of Relation

For a relation R the inverse of the relation is given by R-1 . R-1 = { (y, x) (x, y) ∈ R }

**Types of relations

**Closure

The **closure of a relation refers to the smallest relation that contains the original relation and satisfies a specific property (such as reflexivity, symmetry, or transitivity).

**Equivalence relation

An Equivalence relation is a relation which is reflexive ****,** symmetric, and transitive.
**Example: The relation "is equal to" (=) is an equivalence relation on the set of real numbers (R).

**Equivalence Class : For an equivalence relation R on a set A. For any element a ∈ A, the **equivalence class of a, denoted as [a], is defined as: [a]={x∈A∣(a,x)∈R}
This means [a] contains all elements of A that are related to a under the relation R.

For any two equivalence class either **[a] U [b] = ϕ or [a] = [b]

**Partial Order Relation: A **partial order relation is a binary relation defined on a set that is reflexive antisymmetric and transitive.

**POSET: A POSET (partially ordered set) is a set of itemsthat follows a partial order relations it is represented as [A: R] where A is the set in which the partial order relation R is defined. **Example : ( R, ≤ ) , relation defined by less than equal to on the real numbers.

Hasse Diagram : A **Hasse diagram is a graphical representation of a poset (partially ordered set) that shows the ordering of elements in a simplified way.**

**Minimal and Maximal Elements: In a poset (P,≤), **minimal and **maximal elements are defined as follows:

  1. **Minimal Element:
    • An element a ∈ P is minimal if there is no x ∈ P such that x<a (strictly smaller).
    • a has no smaller element in the poset.
  2. **Maximal Element:
    • An element a∈P is maximal if there is no x∈P such that a<x (strictly larger).
    • a has no larger element in the poset.

**Note: There can be multiple minimal or maximal elements in a poset.

**Least Upper Bound (LUB) : The **least upper bound of a subset S of a poset P is the smallest element in P that is **greater than or equal to every element in S. It is also known as supremum and is denoted as sup(S). If u is an upper bound: For all x ∈ S, x ≤ u.

**Greatest Lower Bound (GLB) : The **greatest lower bound of a subset S of a poset P is the largest element in P that is **less than or equal to every element in S. It is also known as Infimum and Denoted as inf(S). If g is a lower bound: For all x ∈ S, g ≤ x.

Lattice

A**latticeis a special type of partially ordered set (**poset) in which **every pair of elements has both **Least Upper Bound (LUB) and Greatest Lower Bound (GLB).

**Types of lattice :

**Groups

A **group is a set G with a binary operation + that satisfies:

  1. **Closure: For all a, b ∈ G a + b ∈ G.
  2. **Associativity: (a + b) + c = a + (b + c) for all a, b, c ∈ G.
  3. **Identity Element: There exists an element e ∈ G such that a + e = e + a = a for all a ∈ G.
  4. **Inverse Element: For every a ∈ G, there exists b ∈ G such that a + b = b + a = e.

**Semigroup: A **semigroup is a set S with a binary operation + that satisfies the conditions for closure and Associativity .

**Monoid : A **Monoid is a set S with a binary operation + that satisfies the conditions for closure and Associativity and has an identity element.

**Subgroup : A **subgroup is a subset H of a group G that itself forms a group under the same binary operation. Intersection of two subgroup is also a subgroup but the union of two subgroups does not necessarily a sub group

**Cyclic Group : A **cyclic group is a group G that can be generated by a single element g, Every element of G can be written as gn for some integer n.
All sub group of a cyclic group are cyclic.

**Read more : **Group Theory

Graph Theory