Math by Tathagata
Kuratowski’s first graph usually refers to the non-planar complete graph on five vertices denoted by K5 whereas Kuratowski’s second graph K3,3 is the complete bipartite graph with two sets of 3 vertices which is also non-planar. Lets discuss about Kuratowski first and second graph with their properties. What is Kuratwoski’s First Graph? Definition: Kuratowski’s first … Read more
n this page, let us prove a graph is a tree if and only if it is minimally connected. Before proving this, let us recall the definition of a tree. Definition of a Tree in Graph Theory A connected graph without any circuit is called a tree. Tree is Minimally Connected Proof Theorem: Show that … Read more
In this page, let us prove chromatic number of a cycle with n vertices is 2 if n even and 3 if n odd. Before proving this, let us recall the definition of the chromatic number of a graph. Chromatic Number of a Graph Definition: The chromatic number of a graph is the minimum number of … Read more
Discrete Math Problems: In this page, you can find a list of Discrete Math Problems which can be useful for practice for final exam. Discrete Math Practice Problems Q1: Without using truth table show the following: [Solution] Q2: Q3: State the inclusion-exclusion principle for two/three sets. Among 100 patients admitted to a hospital, 50 are … Read more
In this page, let us prove that the chromatic number of a complete graph with n vertices is n. Before proving this, let us recall the definition of the chromatic number of a graph. Chromatic Number of a Graph Definition: The chromatic number of a graph G, denoted by $\chi(G)$, is the minimum number of … Read more
In this page we will prove that a distributive lattice is modular. Distributive Lattice: A lattice (L, ≤) is called distributive if the distributive property holds, i.e,a∨(b∧c) = (a∨b)∧(a∨c)a∧(b∨c) = (a∧b)∨(a∧c) Modular Lattice: A lattice (L, ≤) is called modular if for all a, b, c ∈ L with a≤c we have (a∨b)∧c = a∨(b∧c). … Read more
Conjunctive and Disjunctive Normal Form: Product of elementary sums are called conjunctive normal form (CNF) whereas sum of elementary products are called disjunctive normal form (DNF). Elementary product: Product of the variables and their negations. For example, p∧q, p∧~q, ~p∧~q, p, q etc. Elementary sum: Sum of the variables and their negations. For example, p∨q, … Read more
Two propositions are logically equivalent if they always have the same truth value in every scenario. In this page, we will study about logical equivalence with their laws, truth table and examples. Definition of Logical Equivalence Two propositions p and q are called logically equivalent if both statements have the same truth table or the … Read more
Tautology in Logic: Tautology, contradiction, and contingency classify propositional statements by their truth values: a tautology is always true, a contradiction is always false, and a contingency can be either true or false depending on variable assignments. These concepts are determined using truth tables, examining if the final column shows all True (tautology), all False … Read more
Lets learn about Mathematical Logic in this page. Proposition A declarative statement (or assertion) that has a definite truth value, either true or false, but not both, is called a proposition. Lower case letters are often used to denote propositions, for example, p, q, r etc. NOTE: Examples of Propositions The following are few examples … Read more