Chapter 3 Direct Proof and Proof by Contrapositive Flashcards
Axiom
Axiom - A true mathematical statement whose truth is accepted without proof is referred to as an axiom. In other words, A statement that is taken to be true, so that further reasoning can be done. It is not something we want to prove.
Ex. a + b = b + a for any two numbers a and b.
Theorem
Theorem - A true mathematical statement whose truth can be verified is often referred to as a theorem, although many mathematicians reserve the work “theorem” for such statements that are especially significant or interesting.In other words, A result that has been proved to be true (using operations and facts that were already known).
Ex: The “Pythagoras Theorem” proved that a^2 + b^2 = c^2 for a right angled triangle.
Intermediate Value Theorem
- Binomial Theorem
- Fundamental Theorem of Arithmetic
- Fundamental Theorem of Algebra
Corollary
Corollary - A corollary is a mathematical result that can be deduced from, and is thereby a consequence of, some earlier result. In other words, it is a theorem that follows on from another theorem.
Ex. there is a Theorem that says: two angles that together form a straight line are “supplementary” (they add to 180°). A Corollary to this is the “Vertical Angle Theorem” that says: where two lines intersect, the angles opposite each other are equal (a=c and b=d in the diagram).
Lemma
Lemma - A lemma is a mathematical result that is useful in establishing the truth of some other result. In other words, Like a Theorem, but not as important. It is a minor result that has been proved to be true (using facts that were already known).
Ex. For all real numbers r, |-r| = r
Direct proof
Divides Defination
Even Defination
Odd Defination
Contrapositive Def.
Proof by Contrapositive -
Proof By Cases
Of the same parity Vs of the opposite parity
