Proof meaning nature & its types

about proof in mathematics

1. K.L.E.SOCIETY’S, COLLEGE OF EDUCATION VIDYANAGAR, HUBBALLI-31 POSS MATHEMATICS TOPIC : PROOF-MEANING , NATURE , TYPES submitted to submitted by Shri. VEERESH .A. KALAKERI Mr.R. PRAKASH BABU LECTURER B.Ed ., 1 st Semester

2. PROOF- MEANING & NATURE MEANING : The word “proof “comes from the Latin word “probare” means “to test“ NATURE : Aesthetic, logical deductive arguments for mathematical statements A rigorous argument to convince the truth of a statement Proofs are “theorems” & “axioms”

3. DIRECT PROOF – TYPES & EXAMPLES Conclusion is established logically. Used to prove arithmetic sums,algebraic expressions & geometrical theorems Eg., The sum of two even integer Is always even TYPES : 1. EXPERIMENTAL PROOF Theoretical argument is replaced by practical process. Logic from known to unknown( Deductive approach) Eg., The vertical opposite angles are equal.

4. Examples of experimental proof

5. 2. LOGICAL PROOF An abstract argument with four steps, The data, to prove,construction & proof This is from unknown to known (synthetic approach) Eg., PYTHAGORAS THEOREM 3. INTUITIVE PROOF Not applicable for all theorems No need of experimental or logical proof It needs figures & measurements Eg ., Parallel lines never meet each other

6. INDIRECT PROOF- TYPES &EXAMPLES Used to prove proposition which cannot be proved by direct proof Looks for a contradiction to its original assumption TYPES : 1. PROOF BY CONTRADICTION : Starts the stated conclusion assuming to be untrue,for the same of reasoning & reaches absurdity to infer that the assumption is wrong or untrueEg .,Two adjacent angles are supplementary , then they are in the same straight line

7. 2. PROOF BY EXHAUSTION Conclusion is established by dividing into many number of cases proving each separately Possibility of exhaustion shown by contrary One possible left is accepted. E.g., If two angles of a triangle are unequal, the greater angle has the greater side opposite to it 3. NON-CONSTRUCTIVE PROOF A proof by contradiction in which the non-existence of the object is proved to be impossible. E.g., There exists two irrational numbers a & b such that is a rational number