Chapter 2: Proofs

Content locked

The catalog is public. Reading this content requires approved access.

Chapter 2: Proofs

2.1 Mathematical definitions

Even and odd integers

Check

Show answer

Check

Show answer

Check

Show answer

Rational numbers

PARTICIPATION ACTIVITY

Check

Show answer

Divides

Prime and composite numbers

Definition 2.1.4: Prime and composite numbers.

PARTICIPATION ACTIVITY

2.1.4: Prime and composite numbers.

Inequalities

PARTICIPATION ACTIVITY

2.1.5: Negating inequalities.

Animation content:

Static figure:

Animation captions:

PARTICIPATION ACTIVITY

CHALLENGE ACTIVITY

Additional exercises

EXERCISE

2.1.4: Listing the positive divisors of an integer.

EXERCISE

2.1.5: Prime and composite numbers.

2.2 Introduction to proofs

Animation content:

Animation captions:

Theorems that are universal or existential statements

PARTICIPATION ACTIVITY

Proofs of universal statements: proofs by exhaustion

Proof.

PARTICIPATION ACTIVITY

Proofs of universal statements: universal generalization

Proof.

Counterexamples

PARTICIPATION ACTIVITY

Counterexamples for conditional statements

PARTICIPATION ACTIVITY

Proving existential statements

Disproving existential statements

PARTICIPATION ACTIVITY

CHALLENGE ACTIVITY

Start

EXERCISE

2.3 Best practices and common errors in proofs

Proof steps and assumptions

The rules of algebra.

The set of integers is closed under addition, multiplication, and subtraction.

Every integer is either even or odd.

If x is an integer, there is no integer between x and x + 1 .

The relative order of any two real numbers.

The square of any real number is greater than or equal to 0.

The language of proofs

Figure 2.3.2: Common keywords and phrases in proofs.

Thus and therefore

Let

Suppose

Since

We will prove … and We will show …

By definition

By assumption

In other words

gives and yields

PARTICIPATION ACTIVITY

Proof.

Best practices in writing proofs

Figure 2.3.3: Best practices in writing proofs.

Indicate when the proof starts and ends.

Write proofs in complete sentences.

Give the reader a roadmap of what has been shown, what is assumed, and where the proof is going.

Introduce each variable when the variable is used for the first time.

A block of equations should be introduced with English text and each step that does not follow from algebra should be justified. ©zyBooks 04/08/25 22:07 2501496

PARTICIPATION ACTIVITY

2.3.3: Proof example using best practices.

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

4) Proof.

Existential instantiation

PARTICIPATION ACTIVITY

Common mistakes in proofs

Common mistake 2.3.1: Generalizing from examples.

Common mistake 2.3.2: Skipping steps.

Common mistake 2.3.3: Circular reasoning.

Common mistake 2.3.4: Assuming facts that have not yet been proven.

Additional exercises

EXERCISE

2.4 Writing direct proofs

The form of direct proofs

Reset

Writing direct proofs

Animation content:

Animation captions:

Unused blocks

Proof

Move blc

The process of writing proofs

Proof.

PARTICIPATION ACTIVITY

Unused blocks

Proof

Move blc

A direct proof about rational numbers

Animation content:

Animation captions:

Unused blocks

Proof

Move blc

CHALLENGE ACTIVITY

Unused blocks

Proof

Check

CHALLENGE ACTIVITY

Unused blocks

Proof

Move blc

Check

Start

Additional exercises

EXERCISE

2.5 Proof by contrapositive

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

Proof.

PARTICIPATION ACTIVITY

Unused blocks

Proof

Move blc

Proof.

PARTICIPATION ACTIVITY

Irrational numbers and proofs by contrapositive

PARTICIPATION ACTIVITY

Unused blocks

Proof

Move blc

Proofs by contrapositive of conditional statements with multiple hypotheses

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Start

CHALLENGE ACTIVITY

Unused blocks

Proof

Check

Move blc

CHALLENGE ACTIVITY

Unused blocks

Proof

Move blc

Check

Additional exercises

EXERCISE

2.6 Proof by contradiction

proof.

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

PARTICIPATION ACTIVITY

Proof by contradiction and proof by contrapositive

PARTICIPATION ACTIVITY

A classic proof by contradiction

Proof.

Unused blocks

Proof

More examples of proof by contradiction

Proof.

PARTICIPATION ACTIVITY

Proof.

PARTICIPATION ACTIVITY

Unused blocks

Proof

Check

Unused blocks

Proof

Check

CHALLENGE ACTIVITY

Start

Additional exercises

EXERCISE

2.7 Proof by cases

PARTICIPATION ACTIVITY

Animation content:

Animation captions:

Proof 2.7.1: Proof by cases: mutual friends and enemies.

Proof.

Unused blocks

Proof

Proof by cases and the absolute value function

PARTICIPATION ACTIVITY

2.7.5: Proof by cases: absolute value.

PARTICIPATION ACTIVITY

Full screen

Unused blocks

Proof

Move blc

Without loss of generality

Proof.

PARTICIPATION ACTIVITY

Proof.

Unused blocks

Proof

Check

CHALLENGE ACTIVITY

Unused blocks

Proof

CHALLENGE ACTIVITY

Start

Additional exercises

EXERCISE

Prove each statement.

EXERCISE

Prove each statement.

Source