In this article, you will learn simple tips and tricks to ace math exams on the fly. Although there are no shortcuts to success, focused study, regular practice of solving examples and worksheets, mastering the concepts are some of the best practices."If I were again beginning my studies, I would follow the advice of Plato and start with mathematics." Galileo GalileiStay Focused and Never Give-upWhen you study math, find a quiet place, get rid of the distractions, and focus on your work. You can easily make a mistake or miss a number otherwise.Understand and Master the ConceptsWhichever topic you are ... Read More

Amdahl’s LawSuppose, Moni have to attend an invitation. Moni’s another two friend Diya and Hena are also invited. There are conditions that all three friends have to go there separately and all of them have to be present at door to get into the hall. Now Moni is coming by car, Diya by bus and Hena is coming by foot. Now, how fast Moni and Diya can reach there it doesn’t matter, they have to wait for Hena. So to speed up the overall process, we need to concentrate on the performance of Hena other than Moni or Diya.This is ... Read More

A covering graph is a subgraph that contains either all the vertices or all the edges corresponding to some other graph. A subgraph that contains all the vertices is called a line/edge covering. A subgraph that contains all the edges is called a vertex covering.Let 'G' = (V, E) be a graph. A subset K of V is called a vertex covering of 'G', if every edge of 'G' is incident with or covered by a vertex in 'K'.ExampleTake a look at the following graph −The subgraphs that can be derived from the above graph are as follows −K1 = ... 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 set which contains infinite number of elements is called an infinite set.Example − S = { x | x ∈ N and x > 10 }SubsetA set X is a subset of set Y (Written as X ⊆ Y) if every element of X is an element of set Y.Example ... 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 relation R is defined as − R' = { (b, a) | (a, b) ∈ R }Example − If R = { (1, 2), (2, 3) } then R' will be { (2, 1), (3, 2) }A relation R on set A is called Reflexive if ∀ a ∈ 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 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