Course outline
Module 1
Module 1: Introduction to Mathematical Proofs
Introduction to Mathematical Proofs
Knowledge check
Q1. What is the primary goal of a mathematical proof?
- A. To calculate numerical values
- B. To establish the truth of a mathematical statement ✓
- C. To create visual representations
- D. To solve equations
Why: A mathematical proof aims to logically demonstrate that a statement is true under given conditions.
Q2. Which of the following is a valid form of proof?
- A. Guessing
- B. Intuition
- C. Direct proof ✓
- D. Assumption
Why: A direct proof is a standard method used to prove mathematical statements by logically deducing them from known facts or axioms.
Q3. What does it mean for a statement to be vacuously true?
- A. It is always true
- B. It is true because the premise is false ✓
- C. It is true by definition
- D. It is true in all cases
Why: A vacuously true statement is true because the condition under which it would be false never occurs, often in implications where the antecedent is false.
Q4. What is a lemma in the context of proofs?
- A. A theorem that is not important
- B. A statement used to prove another statement ✓
- C. A conclusion of a proof
- D. A type of mathematical equation
Why: A lemma is a smaller, auxiliary statement that is proven to help in the proof of a larger theorem.
Q5. Which of the following is an example of a proof by contradiction?
- A. Proving that the sum of two even numbers is even
- B. Assuming the opposite of what you want to prove and showing it leads to a contradiction ✓
- C. Using examples to support a claim
- D. Using a diagram to illustrate a point
Why: Proof by contradiction involves assuming the negation of the statement and showing that this assumption leads to a contradiction.
Module 2
Module 2: Logical Reasoning and Propositional Logic
Logical Reasoning and Propositional Logic
Knowledge check
Q1. What is the negation of the proposition "All birds can fly"?
- A. No birds can fly
- B. Some birds cannot fly ✓
- C. All birds cannot fly
- D. Some birds can fly
Why: The negation of a universal statement is an existential statement that asserts the opposite for at least one case.
Q2. Which of the following is a logical implication of the statement "If it rains, then the ground gets wet"?
- A. If the ground is wet, then it rains
- B. If it does not rain, then the ground does not get wet ✓
- C. The ground gets wet if and only if it rains
- D. It rains whenever the ground gets wet
Why: The contrapositive of a conditional statement is logically equivalent to the original statement.
Q3. Which of the following is a conjunction?
- A. Either it is sunny or it is raining
- B. It is sunny and it is raining ✓
- C. If it is sunny, then it is raining
- D. It is not sunny
Why: A conjunction is a logical operation that combines two propositions with "and."
Q4. Which of the following is a disjunction?
- A. Either it is sunny or it is raining ✓
- B. It is sunny and it is raining
- C. If it is sunny, then it is raining
- D. It is not sunny
Why: A disjunction is a logical operation that combines two propositions with "or."
Q5. What is the truth value of the statement "2 + 2 = 4 or 2 + 2 = 5"?
- A. True ✓
- B. False
- C. Unknown
- D. Depends on the context
Why: A disjunction is true if at least one of the propositions is true.
Module 3
Module 3: Direct Proofs and Contrapositive
Direct Proofs and Contrapositive
Knowledge check
Q1. What is the first step in a direct proof?
- A. Assume the conclusion is false
- B. Start with the given premises ✓
- C. Use a contradiction
- D. Prove the contrapositive
Why: A direct proof starts by assuming the premises are true and logically deducing the conclusion.
Q2. Which of the following is a valid statement for a direct proof?
- A. If P then Q. Assume Q is false.
- B. If P then Q. Assume P is true. ✓
- C. If P then Q. Assume not P.
- D. If P then Q. Assume Q is true.
Why: In a direct proof, you assume the premise P is true and show that Q logically follows.
Q3. Which of the following is logically equivalent to the contrapositive of a statement?
- A. The converse
- B. The inverse
- C. The original statement ✓
- D. The negation
Why: The contrapositive of a statement is logically equivalent to the original statement.
Q4. Which of the following is the contrapositive of the statement: If it rains, then the ground is wet?
- A. If the ground is wet, then it rains.
- B. If it does not rain, then the ground is not wet.
- C. If the ground is not wet, then it does not rain. ✓
- D. If the ground is dry, then it does not rain.
Why: The contrapositive of "If P then Q" is "If not Q then not P."
Module 4
Module 4: Proof by Contradiction
Proof by Contradiction
Knowledge check
Q1. What is the first step in a proof by contradiction?
- A. Assume the conclusion is true
- B. Assume the opposite of what you want to prove ✓
- C. Prove the statement directly
- D. Use a known theorem
Why: In proof by contradiction, you start by assuming the opposite of the statement you want to prove.
Q2. Which of the following is a common use of proof by contradiction?
- A. Proving that the square root of 2 is irrational ✓
- B. Proving that 1+1=2
- C. Proving that a triangle has three sides
- D. Proving that a rectangle has four sides
Why: Proof by contradiction is often used to prove statements that are not immediately obvious, such as the irrationality of the square root of 2.
Q3. In a proof by contradiction, after assuming the opposite, what is the next logical step?
- A. Ignore the assumption and proceed with the proof
- B. Derive a contradiction from the assumption ✓
- C. Prove the assumption is true
- D. Conclude the original statement is false
Why: After assuming the opposite, you must derive a contradiction from that assumption to show it is false.
Q4. If you assume that there are finitely many prime numbers and then derive a contradiction, what does this prove?
- A. There are infinitely many prime numbers ✓
- B. There are finitely many prime numbers
- C. Prime numbers are not important
- D. All numbers are prime
Why: Assuming finitely many primes and deriving a contradiction proves that there are infinitely many primes.
Q5. What does it mean if a proof by contradiction leads to a contradiction?
- A. The assumption is true
- B. The assumption is false ✓
- C. The proof is invalid
- D. The statement is true
Why: A contradiction implies that the initial assumption is false, thereby proving the original statement.
Module 5
Module 5: Mathematical Induction
Mathematical Induction
Knowledge check
Q1. What is the first step in a proof by mathematical induction?
- A. Prove the statement is true for all integers
- B. Prove the statement is true for the base case ✓
- C. Assume the statement is true for some arbitrary integer
- D. Prove the statement is true for the next integer
Why: The base case is the first step in induction, where the statement is verified for a specific starting value, typically n = 1.
Q2. In mathematical induction, what is the assumption that the statement is true for some arbitrary integer called?
- A. Base case
- B. Inductive hypothesis ✓
- C. Inductive step
- D. Conclusion
Why: The assumption that the statement is true for some arbitrary integer is called the inductive hypothesis.
Q3. Which of the following is typically used to prove the inductive step in mathematical induction?
- A. Direct proof
- B. Proof by contradiction
- C. Assuming the inductive hypothesis ✓
- D. All of the above
Why: The inductive step usually involves assuming the inductive hypothesis and then proving the statement for the next integer.
Q4. Which of the following is a common base case for mathematical induction?
- A. n = 0
- B. n = 1
- C. n = 2
- D. All of the above ✓
Why: The base case can be any integer depending on the problem, but n = 0, n = 1, and n = 2 are all commonly used.
Q5. What is the final step in a proof by mathematical induction?
- A. Prove the base case
- B. Assume the inductive hypothesis
- C. Prove the inductive step
- D. Conclude the statement is true for all integers greater than or equal to the base case ✓
Why: After proving the base case and the inductive step, the conclusion is that the statement is true for all integers greater than or equal to the base case.
Module 6
Module 6: Set Theory and Proofs
Set Theory and Proofs
Knowledge check
Q1. What is the union of the sets {1, 2, 3} and {2, 3, 4}
- A. {1, 2, 3, 4} ✓
- B. {2, 3}
- C. {1, 4}
- D. {1, 2, 3, 4, 5}
Why: The union of two sets contains all elements from both sets without duplication.
Q2. Which of the following is the correct notation for the empty set?
- A. { }
- B. ( )
- C. []
- D. ∅ ✓
Why: The symbol ∅ is the standard notation for the empty set in set theory.
Q3. If A = {a, b, c} and B = {c, d, e}, what is A ∩ B?
- A. {a, b, c, d, e}
- B. {c} ✓
- C. {a, b}
- D. {d, e}
Why: The intersection of two sets contains only the elements that are common to both sets.
Q4. Which of the following is a subset of every set?
- A. The universal set
- B. The empty set ✓
- C. The set itself
- D. A proper subset
Why: The empty set is a subset of every set, including itself.
Q5. What is the Cartesian product of {1, 2} and {a, b}?
- A. {(1, a), (1, b), (2, a), (2, b)} ✓
- B. {(1, a), (2, b)}
- C. {1, 2, a, b}
- D. {(1, 2), (a, b)}
Why: The Cartesian product of two sets is the set of all ordered pairs where the first element is from the first set and the second is from the second set.
Module 7
Module 7: Common Proof Techniques in Algebra
Common Proof Techniques in Algebra
Knowledge check
Q1. What is the first step in proving a statement by mathematical induction?
- A. Assume the statement is true for all integers
- B. Prove the base case is true ✓
- C. Assume the statement is true for some integer k
- D. Prove the statement is true for k+1
Why: The base case is the first step in induction as it establishes the initial truth of the statement.
Q2. Which of the following is a valid method to prove that two sets are equal?
- A. Show that one is a subset of the other
- B. Show that both sets have the same number of elements
- C. Show that one set is a proper subset of the other
- D. Show that both sets are subsets of each other ✓
Why: To prove two sets are equal, you must show that each is a subset of the other.
Q3. Which of the following is a direct proof technique?
- A. Proof by contradiction
- B. Proof by contrapositive ✓
- C. Proof by induction
- D. Proof by example
Why: Proof by contrapositive is a direct method that involves proving the contrapositive of the original statement.
Q4. What is the contrapositive of the statement "If it rains, then the ground is wet"?
- A. If the ground is wet, then it rains
- B. If it does not rain, then the ground is not wet
- C. If the ground is not wet, then it does not rain ✓
- D. If it rains, then the ground is not wet
Why: The contrapositive swaps and negates both the hypothesis and the conclusion.
Q5. Which of the following is a common proof technique used to show that a function is injective?
- A. Proof by contradiction
- B. Proof by contrapositive ✓
- C. Proof by induction
- D. Proof by counterexample
Why: Proof by contrapositive is often used to show injectivity by assuming the conclusion is false and showing the hypothesis must also be false.