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## Shah Alam

I am a student of mathematics department of Begum Rokeya University, Rangpur. I want to provide all kinds of solution of mathematics problems by this web site.

## 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 ...

## Absorption Property, Idempotent Property| Discrete mathematics

Absorption Property: For any a, b in A, a˅(a˄b) =a and a˄(a˅b)=a Proof: Since a˅(a˄b) is the join of a and (a˄b) then , a˅(a˄b) ≥a ————————-(1) Since a≥a and a≥(a˄b) which implies that   (a˅a) ≥ a˅(a˄b) ⇒ a≥ a˅(a˄b) ———————————(2) From (1) and (2) we get, a˅(a˄b)=a Now ...

## Prove that the line segment joining the middle point of the hypotenuse of a right angle triangle and the opposite vertex is half the hypotenuse.

General enunciation: We have to prove that the line segment joining the middle point of the hypotenuse of a right angle triangle and the opposite vertex is half the hypotenuse. Particular enunciation: Suppose in right angle triangle ∆ABC, ∠B=900 and AC is hypotenuse. BO is the line joining the middle ...

## Both the join and meet operations are associative

Both the join and meet operations are associative i.e., a ˅(b˅c) =(a˅b) ˅c and a˄(b˄c)=(a˄b) ˄c Proof: We first show that, the join operation is associative i.e., a ˅(b˅c) =(a˅b) ˅c Let a ˅(b˅c) = g and (a˅b) ˅c=h Since g is the least upper bound of a and (b˅c) ...

## Echelon matrices, Row canonical form, Row equivalence

Echelon matrices: A matrix A is called an echelon matrix, or is said to be in echelon form, if the following two conditions: (1) All zero rows, if any, are at the bottom of the matrix. (2) Each leading non-zero entry in a row is to the right of leading ...

## ABC is an isosceles triangle and AB=AC. The side BC is extend up to D. Prove that AD>AB.

General enunciation: ABC is an isosceles triangle and AB=AC. The side BC is extend up to D. Prove that AD>AB. Particular enunciation: Given that, ABC is an isosceles triangle and AB=AC. The side BC is extend up to D. It is required to prove that AD>AB. Proof: In ∆ ABC ...

## Show that R is a partially ordering relation.

For a given set A, consider the relation R={(x,y)|x∊P(A), y∊P(A) and x≤y. Show that R is a partially ordering relation. Proof: For all a∊P(A), a≤a ⟹(a,a) ∊R, R is reflexive Let (a,b) ∊R and (b,a) ∊R then a≤b and b≤a implies that a=b. ∴ R is an anti-symmetric. Let (a,b) ...