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 ...

Read More »## 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 ...

Read More »## 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 ...

Read More »## 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 ...

Read More »## 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) ...

Read More »## 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 ...

Read More »## 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 ...

Read More »## 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) ...

Read More »## Let R be a reflexive relation on a set A. Show that R is an equivalence relation if and only if (a,b) and (a,c) in R implies that (b,c) ∊R.

Proof: First we say that R is an equivalence relation. We have (a,b)∊R and (a,c) ∊R ⟹(b,a) ∊R and (a,c) ∊R [∵ R is symmetric] ⟹(b,c) ∊R Conversely, we suppose that (b,c) ∊R Given that R is reflexive. Let (a,b) ∊R, then there exist x∊A such that (x,a) ∊R and ...

Read More »## Profit and loss

Definition of cost price and selling price: The purchasing price of anything is called cost price and the selling price of that thing is called selling price. Or, The expenditure which is done for buying or manufacturing commodities is the cost price and the price that is obtained by selling ...

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