Calculus| Limit:Calculus – limitRead More »
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 ...Read More »
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 ...Read More »
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 ...Read More »
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 ″ ...Read More »
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 ...Read More »
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 »
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: 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 »