Power set of a set S is the set of all subsets of S including the empty set. The cardinality of a power set of a set S of cardinality n is 2n. Power set is denoted as P(S).Example −For a set S = { a, b, c, d } let us calculate the subsets −Subsets with 0 elements − { ∅ } (the empty set)Subsets with 1 element − { a }, { b }, { c }, { d }Subsets with 2 elements − { a, b }, { a, c }, { a, d }, { b, ... Read More

Planar graph − A graph G is called a planar graph if it can be drawn in a plane without any edges crossed. If we draw graph in the plane without edge crossing, it is called embedding the graph in the plane.Non-planar graph − A graph is non-planar if it cannot be drawn in a plane without graph edges crossing.

A graph 'G' is said to be planar if it can be drawn on a plane or a sphere so that no two edges cross each other at a non-vertex point.ExampleRegionsEvery planar graph divides the plane into connected areas called regions.ExampleDegree of a bounded region r = deg(r) = Number of edges enclosing the regions r.deg(1) = 3 deg(2) = 4 deg(3) = 4 deg(4) = 3 deg(5) = 8Degree of an unbounded region r = deg(r) = Number of edges enclosing the regions r.deg(R1) = 4 deg(R2) = 6In planar graphs, the following properties hold good −1. In a ... Read More

Pendent VertexBy using degree of a vertex, we have a two special types of vertices. A vertex with degree one is called a pendent vertex.ExampleHere, in this example, vertex 'a' and vertex 'b' have a connected edge 'ab'. So with respect to the vertex 'a', there is only one edge towards vertex 'b' and similarly with respect to the vertex 'b', there is only one edge towards vertex 'a'. Finally, vertex 'a' and vertex 'b' has degree as one which are also called as the pendent vertex.Isolated VertexA vertex with degree zero is called an isolated vertex.ExampleHere, the vertex 'a' ... Read More

Partition of a set, say S, is a collection of n disjoint subsets, say P1, P1, ... Pn that satisfies the following three conditions −Pi does not contain the empty set.                        [ Pi ≠ { ∅ } for all 0 < i ≤ n ]The union of the subsets must equal the entire original set.                       [ P1 ∪ P2 ∪ ... ∪ Pn = S ]The intersection of any two distinct sets is empty.            ... Read More

A spanning tree with assigned weight less than or equal to the weight of every possible spanning tree of a weighted, connected and undirected graph $G$, it is called minimum spanning tree (MST). The weight of a spanning tree is the sum of all the weights assigned to each edge of the spanning tree. Following are two most popular algorithms to find a minimum spanning tree (MST).Kruskal's AlgorithmKruskal's algorithm is a greedy algorithm that finds a minimum spanning tree for a connected weighted graph. It finds a tree of that graph which includes every vertex and the total weight of ... Read More

A graph can be represented using Adjacency Matrix way.Adjacency MatrixAn Adjacency Matrix A[V][V] is a 2D array of size V × V where $V$ is the number of vertices in a undirected graph. If there is an edge between Vx to Vy then the value of A[Vx][Vy]=1 and A[Vy][Vx]=1, otherwise the value will be zero.For a directed graph, if there is an edge between Vx to Vy, then the value of A[Vx][Vy]=1, otherwise the value will be zero.Adjacency Matrix of an Undirected GraphLet us consider the following undirected graph and construct the adjacency matrix −Adjacency matrix of the above undirected ... Read More

PropositionA proposition is a collection of declarative statements that has either a truth value "true” or a truth value "false". A propositional consists of propositional variables and connectives. We denote the propositional variables by capital letters (A, B, etc). The connectives connect the propositional variables.PredicateA predicate is an expression of one or more variables defined on some specific domain. A predicate with variables can be made a proposition by either assigning a value to the variable or by quantifying the variable.The following are some examples of predicates −Let E(x, y) denote "x = y"Let X(a, b, c) denote "a + ... Read More

Mathematics can be broadly classified into two categories −Continuous Mathematics − It is based upon continuous number line or the real numbers. It is characterized by the fact that between any two numbers, there are almost always an infinite set of numbers. For example, a function in continuous mathematics can be plotted in a smooth curve without breaks.Discrete Mathematics − It involves distinct values; i.e. between any two points, there are a countable number of points. For example, if we have a finite set of objects, the function can be defined as a list of ordered pairs having these objects, ... Read More

TautologiesA Tautology is a formula which is always true for every value of its propositional variables.Example − Prove [ (A → B) ∧ A ] → B is a tautologyThe truth table is as follows −ABA → B(A → B) ∧ A[ (A → B) ∧ A ] → BTrueTrueTrueTrueTrueTrueFalseFalseFalseTrueFalseTrueTrueFalseTrueFalseFalseTrueFalseTrueAs we can see every value of [ (A → B) ∧ A ] → B is "True", it is a tautology.ContradictionsA Contradiction is a formula which is always false for every value of its propositional variables.Example − Prove (A ∨ B) ∧ [ ( ¬ A) ∧ (¬ B) ] ... Read More