# Last Minute Notes – Discrete Mathematics

See Last Minute Notes on all subjects here.

## Propositional Logic

**Implication( →)**: For any two propositions p and q, the statement “if p then q” is called an implication and it is denoted by p → q.**if and only if(↔)**: For any two propositions p and q, the statement “p if and only if(iff) q” is called a biconditional and it is denoted by p ↔ q.**Closure:**(a*b) belongs to G for all a, b ∈ G.**Associativity:**a*(b*c) = (a*b)*c ∀ a, b, c belongs to G.**Identity Element:**There exists e ∈ G such that a*e = e*a = a ∀ a ∈ G**Inverses:**∀ a ∈ G there exists a^{-1}∈ G such that a*a^{-1}= a^{-1}*a = e- A graph is planar if and only if it does not contain a subdivision of K
_{5}and K_{3, 3}as a subgraph. - Let G be a connected planar graph, and let n, m and f denote, respectively, the numbers of vertices, edges, and faces in a plane drawing of G. Then n – m + f = 2.
- Let G be a connected planar simple graph with n vertices and m edges, and no triangles. Then m ≤ 2n – 4.
- Let G be a connected planar simple graph with n vertices, where n ? 3 and m edges. Then m ≤ 3n – 6.

**De Morgan’s Law **:

**Special Conditional Statements**

1.**Implication :**

2.**Converse :** The converse of the proposition is

3.**Contrapositive :** The contrapositive of the proposition is

4.**Inverse :** The inverse of the proposition is

**Types of propositions based on Truth values**

1.**Tautology** – A proposition which is always true, is called a tautology.

2.**Contradiction** – A proposition which is always false, is called a contradiction.

3.**Contingency** – A proposition that is neither a tautology nor a contradiction is called a contingency.

**There are two very important equivalences involving quantifiers**

1. 2.

**Rules of inference**

## Combinatrics

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

**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.

which gives us-

**Binomial Coefficients**: The -combinations from a set of elements if denoted by . This number is also called a binomial coefficient since it occurs as a coefficient in the expansion of powers of binomial expressions.

Let and be variables and be a non-negative integer. Then

**The binomial expansion using Combinatorial symbols**

## Set Theory

A **Set **is an unordered collection of objects, known as elements or members of the set.

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.

**Equal sets**

Two sets are said to be equal if both have same elements. For example A = {1, 3, 9, 7} and B = {3, 1, 7, 9} are equal sets.

**NOTE: Order of elements of a set doesn’t matter.**

**Subset**

A set A is said to be **subset **of another set B if and only if every element of set A is also a part of other set B.

Denoted by ‘**⊆**‘.

‘A ⊆ B ‘ denotes A is a subset of B.

To prove A is the subset of B, we need to simply show that if x belongs to A then x also belongs to B.

To prove A is not a subset of B, we need to find out one element which is part of set A but not belong to set B.

‘U’ denotes the universal set. Above Venn Diagram shows that A is a subset of B.

**Size of a Set**

Size of a set can be finite or infinite.

For example

Finite set: Set of natural numbers less than 100. Infinite set: Set of real numbers.

Size of the set S is known as **Cardinality number**, denoted as |S|.

Note: Cardinality of a null set is 0.

**Power Sets**

The power set is the set all possible subset of the set S. Denoted by P(S).

Example: What is the power set of {0, 1, 2}?

Solution: All possible subsets

{∅}, {0}, {1}, {2}, {0, 1}, {0, 2}, {1, 2}, {0, 1, 2}.

Note: Empty set and set itself is also the member of this set of subsets.

**Cardinality of power set** is , where n is the number of elements in a set.

**Cartesian Products**

Let A and B be two sets. Cartesian product of A and B is denoted by A × B, is the set of all ordered pairs (a, b), where a belong to A and b belong to B.

A × B = {(a, b) | a ∈ A ∧ b ∈ B}.

**The cardinality of A × B** is N*M, where N is the Cardinality of A and M is the cardinality of B.

Note: A × B is not the same as B × A.

**Union**

Union of the sets A and B, denoted by A ∪ B, is the set of distinct element belongs to set A or set B, or both.

**Intersection**

The intersection of the sets A and B, denoted by A ∩ B, is the set of elements belongs to both A and B i.e. set of the common element in A and B.

**Disjoint**

Two sets are said to be disjoint if their intersection is the empty set .i.e sets have no common elements.

**Set Difference**

Difference between sets is denoted by ‘A – B’, is the set containing elements of set A but not in B. i.e all elements of A except the element of B.**Complement**

The complement of a set A, denoted by , is the set of all the elements except A. Complement of the set A is U – A.

**Formula:**

**Group**

A non-empty set G, (G, *) is called a group if it follows the following axiom:

## Relations And Functions

|A| = m and |B| = n, then

1. No. of functions from A to B = n^{m}

2. No. of one to one function = (n, P, m)

3. No. of onto function =n^{m} – (n, C, 1)*(n-1)^{m} + (n, C, 2)*(n-2)^{m} …. +(-1)^{m}*(n, C, n-1), if m >= n; 0 otherwise

4. Necessary condition for bijective function |A| = |B|

5. The no. of bijection function =n!

6. No. of relations =2^{mn}

7. No. of reflexive relations =2^{n(n-1)}

8. No. of symmetric relations = 2^{n(n+1)/2}

9. No. of Anti Symmetric Relations = 2^{n}*3^{n(n-1)/2}

10. No. of asymmetric relations = 3^{n(n-1)/2}

11. No. of irreflexive relations = 2^{n(n-1)}

12. A relation is a partial order if

1) Reflexive 2) Antisymmetric 3) Transitive

13. Meet Semi Lattice :

For all a, b belongs to L a∧b exists

14. Join Semi Lattice

For all a, b belongs to L a∨b exists

15. A poset is called Lattice if it is both meet and join semi-lattice

16. Complemented Lattice : Every element has complement

17. Distributive Lattice : Every Element has zero or 1 complement .

18. Boolean Lattice: It should be both complemented and distributive. Every element has exactly one complement.

19. A relation is an equivalence if

1) Reflexive 2) symmetric 3) Transitive

## Graph Theory

1. No. of edges in a complete graph = n(n-1)/2

2. Bipartite Graph : There is no edges between any two vertices of same partition . In complete bipartite graph no. of edges =m*n

3. Sum of degree of all vertices is equal to twice the number of edges.

4. Maximum no. of connected components in graph with n vertices = n

5. Minimum number of connected components =

0 (null graph) 1 (not null graph)

6. Minimum no. of edges to have connected graph with n vertices = n-1

7. To guarantee that a graph with n vertices is connected, minimum no. of edges required = {(n-1)*(n-2)/2 } + 1

8. A graph is euler graph if it there exists atmost 2 vertices of odd – degree

9. Tree

-> Has exactly one path btw any two vertices -> not contain cycle -> connected -> no. of edges = n -1

10. For complete graph the no . of spanning tree possible = n^{n-2}

- 11. For simple connected planar graph

- 12.) Every bipartite graph is 2 colourable and vice versa

13.) The no. of perfect matchings for a complete graph (2n)/(2

^{n}n!)

14.) The no. of complete matchings for K

_{n.n}= n!