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Computer Engineering Articles
Page 5 of 35
Mathematical Logical Connectives
A logical connective is a symbol used to connect two or more propositional or predicate logics in such a manner that the resultant logic depends only on the input logics and the meaning of the connective used. There are five fundamental connectives in mathematical logic − OR (∨) − Disjunction AND (∧) − Conjunction NOT (¬) − Negation IF-THEN (→) − Implication IF AND ONLY IF (⇔) − Biconditional OR (∨) − Disjunction The OR operation of two propositions A and B (written as A ∨ B) is true if at least one of ...
Read MoreKirchoff's Theorem
Kirchhoff's theorem (also known as the Matrix Tree Theorem) provides a way to find the number of spanning trees in a connected graph using matrices. Instead of manually listing all spanning trees, this theorem lets you compute the count using the determinant of a special matrix derived from the graph. How Kirchhoff's Theorem Works The process involves three steps − Create the Adjacency Matrix (A) − Fill entry A[i][j] as 1 if there is an edge between vertex i and vertex j, else 0. Create the Degree Matrix (D) − A diagonal matrix where D[i][i] equals ...
Read MoreHamiltonian Graphs
A Hamiltonian graph is a connected graph that contains a cycle which visits every vertex exactly once and returns to the starting vertex. This cycle is called a Hamiltonian cycle. A Hamiltonian path (or walk) passes through each vertex exactly once but does not need to return to the starting vertex. Unlike Eulerian graphs (which require traversing every edge), Hamiltonian graphs focus on visiting every vertex. Sufficient Conditions for Hamiltonian Graphs There is no simple necessary and sufficient condition to determine if a graph is Hamiltonian. However, two important theorems provide sufficient conditions − Dirac's Theorem ...
Read MoreIsomorphism and Homeomorphism of graphs
In graph theory, isomorphism and homomorphism are ways to compare the structure of two graphs. Isomorphism checks whether two graphs are structurally identical, while homomorphism is a more relaxed mapping that preserves adjacency but does not require a one-to-one correspondence. Isomorphism Two graphs G and H are called isomorphic (denoted by G ≅ H) if they contain the same number of vertices connected in the same way. Formally, there must exist a bijective function f: V(G) → V(H) such that two vertices are adjacent in G if and only if their images are adjacent in H. Checking ...
Read MoreCardinality of a Set
The cardinality of a set S, denoted by |S|, is the number of elements in the set. This number is also referred to as the cardinal number. If a set has an infinite number of elements, its cardinality is ∞. Examples of Cardinality |{1, 4, 3, 5}| = 4 (finite set with 4 elements) |{1, 2, 3, 4, 5, ...}| = ∞ (infinite set of natural numbers) |{}| = 0 ...
Read MoreFunctions of Set
A function assigns to each element of a set, exactly one element of a related set. Functions find their application in various fields like representation of the computational complexity of algorithms, counting objects, study of sequences and strings, to name a few. Function − Definition A function or mapping (defined as f: X → Y) is a relationship from elements of one set X to elements of another set Y (X and Y are non-empty sets). X is called the Domain and Y is called the Codomain of function f. Function f is a relation on X ...
Read MoreFinding the number of regions in the graph
In a connected planar graph, the plane is divided into distinct areas called regions (or faces), including the outer unbounded region. The number of regions can be found using Euler's formula for planar graphs, which relates vertices, edges, and regions. Key Formulas Sum of Degrees Theorem − The sum of the degrees of all vertices equals twice the number of edges − ∑ deg(Vi) = 2|E| Euler's Formula − For any connected planar graph − |V| + |R| = |E| + 2 Where |V| is the number of vertices, |E| ...
Read MoreFinding the simple non-isomorphic graphs with n vertices in a graph
Two graphs are isomorphic if one can be transformed into the other by renaming its vertices. In other words, they have the same structure even if the vertices are labeled differently. Non-isomorphic graphs are graphs that have genuinely different structures − no renaming of vertices can make one look like the other. When counting simple non-isomorphic graphs with n vertices, we look for all structurally distinct graphs possible, ignoring vertex labels. Problem Statement How many simple non-isomorphic graphs are possible with 3 vertices? Solution With 3 vertices, there are at most ⌈3C2⌉ = 3 possible ...
Read MoreComposition of Functions of Set
Two functions f: A → B and g: B → C can be composed to give a composition g o f. This is a function from A to C defined by − (g o f)(x) = g(f(x)) In composition, the output of the first function becomes the input of the second function. The function on the right (f) is applied first, and then the function on the left (g) is applied to the result. A B C ...
Read MoreIntroduction to Mathematical Logic
The rules of mathematical logic specify methods of reasoning mathematical statements. Greek philosopher, Aristotle, was the pioneer of logical reasoning. Logical reasoning provides the theoretical base for many areas of mathematics and consequently computer science. It has many practical applications in computer science like design of computing machines, artificial intelligence, definition of data structures for programming languages etc.Major CategoriesMathematical logics can be broadly categorized into three categories.Propositional Logic − Propositional Logic is concerned with statements to which the truth values, "true" and "false", can be assigned. The purpose is to analyse these statements either individually or in a composite manner.Predicate ...
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