What You Need to Know

Key Concept Diagram

A direct proof assumes the hypothesis and uses logical steps to derive the conclusion

Proof by contradiction assumes the statement is false and derives a logical impossibility

Proof by contrapositive proves "if not Q then not P" which is logically equivalent to "if P then Q"

Mathematical induction proves statements for all natural numbers using a base case and inductive step

Key Vocabulary

Axiom

A statement accepted as true without proof, forming the foundation of a mathematical system

Theorem

A statement that has been proven to be true using logical deduction from axioms or previously proven results

Contrapositive

The logically equivalent statement formed by negating and swapping the hypothesis and conclusion

Inductive Step

The part of a proof by induction that shows if the statement holds for n=k, it must also hold for n=k+1

Knowledge Check

Select the correct answer for each question. Click "Check Answer" to see if you are right.

Question 1

In proof by contradiction, what is the starting assumption?

The statement to be proved is true The axioms are false The statement to be proved is false The conclusion is undefined

Question 2

What are the two components required in a proof by mathematical induction?

A contradiction and a direct proof An axiom and a definition A theorem and its converse A base case and an inductive step

Question 3

The contrapositive of "If it is raining, then the ground is wet" is:

If the ground is not wet, then it is not raining If it is not raining, then the ground is not wet If the ground is wet, then it is raining If the ground is wet, then it is not raining

Mathematical Proofs