To deduce new statements from the statements whose truth that we already know, Rules of Inference are used.What are Rules of Inference for?Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements.An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “$\therefore$”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises.Rules of Inference provide the templates or guidelines for constructing valid ... Read More

If G = (V, E) be a non-directed graph with vertices V = {V1, V2, …Vn} thenn ∑ i=1 deg(Vi) = 2|E|Corollary 1If G = (V, E) be a directed graph with vertices V = {V1, V2, …Vn}, thenn ∑ i=1 deg+(Vi) = |E| = n ∑ i=1 deg−(Vi)Corollary 2In any non-directed graph, the number of vertices with Odd degree is Even.Corollary 3In a non-directed graph, if the degree of each vertex is k, thenk|V| = 2|E|Corollary 4In a non-directed graph, if the degree of each vertex is at least k, thenk|V| = 2|E|Corollary 5In a non-directed graph, if the ... Read More

German mathematician G. Cantor introduced the concept of sets. He had defined a set as a collection of definite and distinguishable objects selected by the means of certain rules or descriptions.Set theory forms the basis of several other fields of study like counting theory, relations, graph theory, and finite state machines. In this chapter, we will cover the different aspects of Set Theory.Set - DefinitionA set is an unordered collection of different elements. A set can be written explicitly by listing its elements using a set bracket. If the order of the elements is changed or any element of a ... Read More

Venn diagram, invented in 1880 by John Venn, is a schematic diagram that shows all possible logical relations between different mathematical sets.ExamplesSet OperationsSet Operations include Set Union, Set Intersection, Set Difference, Complement of Set, and Cartesian Product.Set UnionThe union of sets A and B (denoted by A ∪ B) is the set of elements that are in A, in B, or in both A and B. Hence, A ∪ B = { x | x ∈ A OR x ∈ B }.Example − If A = { 10, 11, 12, 13 } and B = { 13, 14, 15 }, ... Read More

To deduce new statements from the statements whose truth that we already know, Rules of Inference are used.What are Rules of Inference for?Mathematical logic is often used for logical proofs. Proofs are valid arguments that determine the truth values of mathematical statements.An argument is a sequence of statements. The last statement is the conclusion and all its preceding statements are called premises (or hypothesis). The symbol “∴”, (read therefore) is placed before the conclusion. A valid argument is one where the conclusion follows from the truth values of the premises.Rules of Inference provide the templates or guidelines for constructing valid ... Read More

Rooted TreeA rooted tree G is a connected acyclic graph with a special node that is called the root of the tree and every edge directly or indirectly originates from the root. An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered. If every internal vertex of a rooted tree has not more than m children, it is called an m-ary tree. If every internal vertex of a rooted tree has exactly m children, it is called a full m-ary tree. If m = 2, the rooted tree is called a binary tree.Binary ... Read More

A relation can be represented using a directed graph.The number of vertices in the graph is equal to the number of elements in the set from which the relation has been defined. For each ordered pair (x, y) in the relation R, there will be a directed edge from the vertex ‘x’ to vertex ‘y’. If there is an ordered pair (x, x), there will be a self- loop on vertex ‘x’.Suppose, there is a relation R = { (1, 1), (1,2), (3, 2) } on set S = { 1, 2, 3 }, it can be represented by the following graph −

There are mainly two ways to represent a graph −Adjacency MatrixAdjacency ListAdjacency 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. And 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 −The ... Read More

Relations may exist between objects of the same set or between objects of two or more sets.Definition and PropertiesA binary relation R from set x to y (written as xRy or R(x, y)) is a subset of the Cartesian product x × y. If the ordered pair of G is reversed, the relation also changes.Generally an n-ary relation R between sets A1, ... ,\ and\ An is a subset of the n-ary product A1 × ... × An. The minimum cardinality of a relation R is Zero and the maximum is n2 in this case.A binary relation R on a ... Read More

Predicate Calculus deals with predicates, which are propositions containing 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.Consider the following statement.Ram is a student.Now consider the above statement in terms of Predicate calculus.Here "is a student" is a predicate and Ram is subject.Let's denote "Ram" as x and "is a student" as a predicate P then we can write the above statement as P(x).Generally a statement expressed by Predicate must have at ... Read More