Published on 29 Sep 2015

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MATH 135 - Lecture 2: Introduction to Proofs

Types of Statements:

● Proposition → a statement that needs to be proven true or demonstrated false by

a valid argument

● Theorem → a significant proposition

● Lemma → a proposition that is often not useful on its own, but is used to prove

another theorem

● Corollary → a proposition that follows almost immediately without much work

from another theorem

● Axiom → a statement that is assumed to be true without proof

Common Mistakes When Dealing with Proofs:

● Multiplying or dividing by a negative number when dealing with inequalities

○ Ex. -5x > 4 → When dividing both sides of the inequality by -5, the new

equation becomes: x < -⅘

○ Note that the inequality sign is flipped.

● Dividing an expression by 0

○ When using variables, it is not always obvious when division by 0 occurs.

○ Ex. a = b

a2 = ab

a2 - b2 = ab - b2

(a + b)(a - b) = b(a - b)

a + b = b

2b = b

2 = 1

○ Note that a = b, therefore (a - b) = 0, so you are unable to divide the

equation by 0 in the 4th step to reach the 5th step.

● Taking the square root of an expression

○ There exists the possibility of both positive and negative roots

Stewart’s Theorem:

Let ABC be a triangle with AB = c, AC = b and BC = a. If P is a point on BC with BP = m, PC = n

and AP = d, then dad + man = bmb +

cnc.

## Document Summary

Math 135 - lecture 2: introduction to proofs. Proposition a statement that needs to be proven true or demonstrated false by a valid argument. Lemma a proposition that is often not useful on its own, but is used to prove another theorem. Corollary a proposition that follows almost immediately without much work from another theorem. Axiom a statement that is assumed to be true without proof. Multiplying or dividing by a negative number when dealing with inequalities. 5x > 4 when dividing both sides of the inequality by -5, the new equation becomes: x < - . Note that the inequality sign is flipped. When using variables, it is not always obvious when division by 0 occurs. Ex. a = b a2 = ab a2 - b2 = ab - b2 (a + b)(a - b) = b(a - b) a + b = b.