1.7 Introduction to Proofs

Question 1

Proofs

Answer 1

A valid argument that establishes the truth of a mathematical statement

Question 2

What is used to prove a statement

Answer 2

Axioms
Definitions
Previously proved results
Rules of inference

Question 3

Axioms

Answer 3

Statements that are assumed to be true

Question 4

Lemma

Answer 4

A less important theorem that is helpful in the proof

Question 5

Corollary

Answer 5

A theorem that can be established directly from a proven theorem

Question 6

Conjecture

Answer 6

A statement that is being proposed to be a true statement.

Question 7

When does a conjecture become a theorem

Answer 7

When proof of a conjecture is found

Question 8

Types of proof

Answer 8

Direct proof
Proof by contraposition
Proof by contradiction
Proof by induction

Question 8

Direct proof

Answer 8

p->q
1.Assume p
2.Use axioms, definitions…
3.Show q

Question 9

For ∀(x)P(x) in direct proof

Answer 9

Let x be an arbitrary value in the domain, show P(x)

Question 10

Steps to prove ∃(x)P(x) (direct proof)

Answer 10

Build, construct, exhibit a value in the domain, verify p(x)

Question 11

Proof by contraposition (contrapositive)

Answer 11

~q->~p
1. Assume ~q
2. Prove ~p
3. Conclude p->q

Question 12

Proof by contradiction

Answer 12

1. Assume the result to be false
2. Deduce a contradiction

Question 13

Proof of equivalence

Answer 13

To prove p↔q, show p->q and q->p

Question 14

Definition of rational numbers

Answer 14

Question 15

Vacuous Proof

Answer 15

p->q can be proven when we know that p is false

Question 16

Properties of odd integers

Answer 16

n = 2k+1
The product and division of two odd integers is odd
The addition and subtraction of two odd integers is even

Question 17

Properties of even integers

Answer 17

n = 2k
The product, division, addition and subtraction of two even integers is even

Question 18

How to prove that sqrt(2) is irrational

Answer 18

Use proof by contradiction
Assume it is rational. Which means sqrt2 = a/b, b does not equal zero, a&b have no common factor(meaning they are in their lowest form)
Find that a and b are both even, meaning they are divisible by 2, hence a contradiction.