Math by Tathagata
Tathagata Form ODE by Eliminating a and b: In this article, let’s learn how to form an ordinary differential equation (ODE) by eliminating a and b which are arbitrary constants. Lets solve some problems. Solved Problems $\boxed{\color{blue}\textbf{Q}1:}$ Eliminate the arbitrary constants a, b and c to from an ordinary differential equation. $\boxed{\color{red}\textbf{Answer:}}$ Part (1) Given y … 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
In this post, we will determine whether the following argument is valid or invalid. Lets consider the following two examples. Question 1: Check whether the argument is valid or not: p∨q, q→r, p→s, ∼s ⇒ r. Answer: We will proceed as follows: 1. p∨q (given) 2. q∨p (commutativity) 3. q→r (given) 4. ∼q∨r (equivalent to … Read more
Lagrange’s theorem states that the order of any subgroup of a finite group divides the order of the group. In this page, we will state and prove Lagrange’s theorem along with some applications and solved problems. Statement The Lagrange’s theorem for finite groups states the following: Let G be a finite group and H be … Read more
In this page, we prove that every subgroup of cyclic group is Cyclic. Every subgroup of a cyclic group is cyclic Proof: Let G be a cyclic group. By definition, there exists an element g ∈ G such that G = <g> = {gk : k∈$\mathbb{Z}$ }. Let H be a subgroup of G. We … Read more
In this page , you will find Analytical Geometry Practice Problems. The problems are listed below. Q1: To what point the origin is to be moved so that one can get rid of the first degree terms from the equation $x^2+xy+2y^2-7x-5y+12=0$. Q2: The gradient of one of the straight lines of $ax^2+2hxy+by^2=0$ is twice that … Read more
In this page, we prove that every group of prime order is cyclic. Prove that Group of Prime Order is Cyclic Answer: Let G be a group of order p where p is a prime number So |G| = p. As p is a prime number, p>1. That is, |G| > 1. So there is … Read more