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Mathematics Articles
Found 203 articles
What is Amdahl's Law?
Amdahl's Law is a fundamental principle in computer science that describes the theoretical maximum speedup achievable when improving part of a system. It demonstrates that the overall performance improvement is limited by the portion of the system that cannot be enhanced. Consider this analogy: Three friends must travel to a party separately but arrive together to enter. One drives a car, another takes the bus, and the third walks. No matter how fast the car and bus arrive, they must wait for the slowest person (the walker). To improve overall arrival time, focus must be on helping the walker, ...
Read MoreMathematical Logical Terms and Definitions
Tautologies A Tautology is a formula which is always true for every value of its propositional variables. Example − Prove [ (A → B) ∧ A ] → B is a tautology The truth table is as follows − A B A → B (A → B) ∧ A [ (A → B) ∧ A ] → B True True True True True True False False False True False True True False True False False True False True As we can see every value ...
Read MoreMatrix Representation of Graphs
A graph can be represented using an Adjacency Matrix, which is a 2D array that stores the connection information between vertices. Adjacency Matrix An Adjacency Matrix A[V][V] is a 2D array of size V × V where V is the number of vertices in the graph. For an undirected graph, if there is an edge between Vx and Vy, then A[Vx][Vy] = 1 and A[Vy][Vx] = 1 (symmetric matrix). For a directed graph, if there is an edge from Vx to Vy, then only A[Vx][Vy] = 1. Otherwise the value is 0. Adjacency Matrix of an Undirected ...
Read MoreTheory of Inference for the Statement Calculus
To deduce new statements from the statements whose truth 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 hypotheses). 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 ...
Read MoreLine/Edge Covering
A covering graph is a subgraph which contains either all the vertices or all the edges corresponding to some other graph. A subgraph which contains all the vertices is called a line/edge covering. A subgraph which contains all the edges is called a vertex covering. Line Covering Let G = (V, E) be a graph. A subset C(E) is called a line covering of G if every vertex of G is incident with at least one edge in C, i.e., deg(V) ≥ 1 ∀ V ∈ G Because each vertex is connected with another vertex by ...
Read MoreTree or Connected acyclic graph
Trees are graphs that do not contain even a single cycle. They represent hierarchical structure in a graphical form. Trees belong to the simplest class of graphs. Despite their simplicity, they have a rich structure. Trees provide a range of useful applications as simple as a family tree to as complex as trees in data structures of computer science. Tree A connected acyclic graph is called a tree. In other words, a connected graph with no cycles is called a tree. The edges of a tree are known as branches. Elements of trees are called their ...
Read MoreTypes of Relations
A relation on a set can have various properties that classify it into different types. Understanding these types is essential for studying equivalence classes, partial orders, and other structures in discrete mathematics. Empty Relation The empty relation between sets X and Y, or on a set E, is the empty set ∅. No element is related to any other element. Full Relation The full relation (or universal relation) between sets X and Y is the entire Cartesian product X × Y. Every element in X is related to every element in Y. Identity Relation The identity relation on set X is ...
Read MoreSum of Degrees of Vertices Theorem
The Sum of Degrees of Vertices Theorem (also known as the Handshaking Lemma) is a fundamental result in graph theory that relates the sum of all vertex degrees to the number of edges in a graph. The Theorem If G = (V, E) is a non-directed graph with vertices V = {V1, V2, …, Vn}, then − ∑i=1n deg(Vi) = 2|E| This is because each edge contributes exactly 2 to the total degree sum − one for each of its endpoints. Example Each edge adds 2 to total degree ...
Read MoreRooted and Binary Tree
A rooted tree G is a connected acyclic graph with a special node called the root, from which every edge directly or indirectly originates. An ordered rooted tree is a rooted tree where the children of each internal vertex are ordered. If every internal vertex has not more than m children, it is called an m-ary tree. If every internal vertex has exactly m children, it is called a full m-ary tree. If m = 2, the rooted tree is called a binary tree. Rooted Tree (root = a) ...
Read MoreRepresentation of Relations using Graph
A relation can be represented visually using a directed graph (digraph). This graphical representation makes it easy to understand which elements are related and in what direction. How to Represent a Relation as a Graph The rules for converting a relation into a directed graph are − The number of vertices equals the number of elements in the set. For each ordered pair (x, y) in the relation R, draw a directed edge from vertex x to vertex y. If there is an ordered pair (x, x), draw a self-loop on vertex x. Example ...
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