CBSE Class 9 Mathematics Proofs in Mathematics Notes Set 13

Statements

A statement is a sentence that is either true or false but not both. Orders, questions and exclamations are not statements. A statement is valid in mathematics only if it is always true or always false; a statement that is sometimes true and sometimes false is ambiguous.

Axioms, Conjectures and Theorems

An axiom is accepted as true without proof. A conjecture is a statement whose truth has not yet been established. A theorem is a statement whose truth has been proved logically from axioms and earlier theorems.

Parts of a Proof

A proof states what is given (hypothesis), what is to be proved (conclusion), any construction, and then a chain of statements each justified by a reason such as an axiom, a definition or an earlier result.

Counter-examples and Negation

A general statement is disproved by a single counter-example. The negation of a statement denies it. Examples support a statement but never prove it, while a counter-example disproves it.

Making a False Statement True

Add a condition that removes the counter-examples. For instance, 'all prime numbers are odd' becomes true as 'all prime numbers greater than 2 are odd', and 'the diagonals of a parallelogram are equal' becomes true for a rectangle.

Solved Example

Prove that n(n + 1) is divisible by 2 for every natural number n.
If n is even, n = 2m and n(n + 1) = 2m(2m + 1). If n is odd, n + 1 = 2r and n(n + 1) = (2r − 1)(2r). In both cases the product is a multiple of 2.