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Home / Tag Archives: math

# Tag Archives: math

## PDF| Calculus| Limit

Calculus| Limit:Calculus – limit

## If D is the middle point of the side BC of ∆ABC, prove that AB+AC>2AD.

General enunciation: If D is the middle point of the side BC of ∆ABC, prove that AB+AC>2AD. Particular enunciation: Suppose, in triangle ∆ABC, D is the mid-point of BC. Let us join (A, D). Let us prove that AB+AC>2AD. Construction: Let us expand AD up to DE such that, AD ...

## Principle of induction| Elementary number theory

Theorem (Principle of induction): Let S be a set of positive integers with the properties (i) 1 belongs to S and (ii) Whenever the integer K is in S then the next integer K+1 must also be in S. Then S is the set of all positive integers. Proof: Let ...

## Linear combination, Spanning set| Linear Algebra

Linear combination: Let V be a vector space over a field K. A vector v in V is a linear combination of vectors u1, u2,—–, um in V if there exist scalars a1, a2, —–, am in K such that v= a1u1+ a2u2+—+ amum Example: (Linear combination in Rn): Suppose ...

## A tree has two vertices of degree 2. One vertex of degree 3 and 3 vertices of degree 4.How many vertices of degree 1 does it have?

Exercise:A tree has 2n vertices of degree 1. 3n vertices of degree 2 and n vertices of degree 3. Determine the number of vertices and edges in the tree. Solution: 2n vertices has total degree = 2n× 1 = 2n 3n       ″      ″    ″      ″   = 3n×2 =6n n       ″     ...

## Prove that if two chords of a circle bisect each other, their point of intersection is the centre of the circle.

General enunciation: If two chords of a circle bisect each other, we have to show that their point of intersection is the centre of the circle. Particular enunciation: Suppose two chords AC and BD of the circle ABCD bisect each other at the point O i.e., OA = OC and ...

## Well ordering principle and Archimedean property

Well ordering principle: Every non empty set S of non-negative integers contains a least element; that there is some integer a in S such that a≤b for all belonging to S. Theorem(Archimedean property): If a and b are any positive integers then there exists a positive integer n such that ...

## Vector Spaces| Chapter 4|linear algebra

Definition: Let V be a nonempty set of vectors with two operations: (i) Vector addition: This assigns to any u,v ∊V, k∊K a sum u+v in V. (ii) Scalar multiplication: This assigns to any u ∊V, k ∊K a product ku ∊V. Then V is called a vector space if ...