Direct Proof
1 hr 38 min 12 Examples
- How to write a proof — understanding terminology, structure, and method of writing proofs
- What are Constructive Proofs and Direct Proofs? And some important definitions
- Apply a constructive claim to verify the statement (Examples #1-2)
- Use a direct proof to show the claim is true (Examples #3-6)
- Justify the following using a direct proof (Example #7-10)
- Demonstrate the claim using a direct argument (Example #11)
- Find a counterexample to disprove the claim (Example #12a-c)
Indirect Proof
1 hr 43 min 12 Examples
- What is proof by contraposition? with Example #1
- Prove using proof by contrapositive (Examples #2-4)
- What is proof by contradiction? (Examples #5-6)
- Show the square root of 2 is irrational using contradiction (Example #7)
- Demonstrate by indirect proof (Examples #8-10)
- Proof of equivalence (Example #11)
- Justify the biconditional statement (Example #12)
Proof By Cases
1 hr 44 min 6 Examples
- Overview of proof by exhaustion with Example #1
- Prove if an integer is not divisible by 3 (Example #2)
- Verify the triangle inequality theorem (Example #4)
- The sum of two integers is even if and only if same parity (Example #5)
- Verify the rational inequality using four cases (Example #5)
- Demonstrate the absolute value inequality by exhaustion (Example #6)
Logic Proofs
1 hr 40 min 11 Examples
- Existential and Uniqueness Proofs (Examples #1-4)
- Use equivalence and inference rules to construct valid arguments (Examples #5-6)
- Translate the argument into symbols and prove (Examples #7-8)
- Verify using logic rules (Examples #9-10)
- Show the argument is valid using existential and universal instantiation (Example #11)
Proof By Induction
1 hr 48 min 10 Examples
- What is the principle of induction? Using the inductive method (Example #1)
- Justify with induction (Examples #2-3)
- Verify the inequality using mathematical induction (Examples #4-5)
- Show divisibility and summation are true by principle of induction (Examples #6-7)
- Validate statements with factorials and multiples are appropriate with induction (Examples #8-9)
- Use the principle of mathematical induction to prove the inequality (Example #10)
Chapter Test
1 hr 14 min 10 Practice Problems
- Proof by cases: If n^2 is a multiple of 3, then n much be a multiple of 3 (Problem #1)
- Disprove by counterexample (Problems #2-3)
- Prove by contraposition: If n^2 is odd, then n is odd (Problem #4)
- Direct proof: The sum of two odd integers is an even integer (Problem #5)
- Direct proof: The sum of three consecutive odd integers is divisible by 3 (Problem #6)
- Prove by induction (Problems #7-8)
- Logic Proofs (Problems #9-10)
