CrudRepository and JPA repository both are the interface of the spring data repository library. Spring data repository reduces the boilerplate code by providing some predefined finders to access the data layer for various persistence layers.JPA repository extends CrudRepository and PagingAndSorting repository. It inherits some finders from crud repository such as ... Read More

Sets can be classified into many types. Some of which are finite, infinite, subset, universal, proper, singleton set, etc.Finite SetA set which contains a definite number of elements is called a finite set.Example − S = { x | x ∈ N and 70 > x > 50 }Infinite SetA ... Read More

The Empty Relation between sets X and Y, or on E, is the empty set ∅The Full Relation between sets X and Y is the set X × YThe Identity Relation on set X is the set { (x, x) | x ∈ X }The Inverse Relation R' of a ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... 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 ... Read More