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Questions and Answers
What is the key step in the proof by mathematical induction?
What is the key step in the proof by mathematical induction?
- Stating the basis step
- Stating the contradiction
- Stating the conclusion
- Stating the inductive step (correct)
What is the purpose of the basis step in a mathematical induction proof?
What is the purpose of the basis step in a mathematical induction proof?
- To show the statement is true for all values
- To show the statement is contradictory
- To show the statement is false for the first value
- To show the statement is true for the first value (correct)
What is the logical structure of a proof by contradiction?
What is the logical structure of a proof by contradiction?
- Assume the statement is false, and show it leads to the desired conclusion
- Assume the statement is true, and show it leads to the desired conclusion
- Assume the statement is true, and derive a contradiction
- Assume the statement is false, and derive a contradiction (correct)
What is the purpose of the conclusion in a mathematical induction proof?
What is the purpose of the conclusion in a mathematical induction proof?
In the proof by contradiction example, what is the role of the statement ¬p?
In the proof by contradiction example, what is the role of the statement ¬p?
What is the role of the statement ¬r in the proof by contradiction example?
What is the role of the statement ¬r in the proof by contradiction example?
Which of the following is a valid logical equivalence?
Which of the following is a valid logical equivalence?
Which of the following statements is the contrapositive of the statement 'p → q'?
Which of the following statements is the contrapositive of the statement 'p → q'?
In a proof by mathematical induction, what is the purpose of the basis step?
In a proof by mathematical induction, what is the purpose of the basis step?
If the sum of the first n positive integers is given by the formula $S_n = \frac{n(n+1)}{2}$, which of the following represents the inductive step in proving this formula using mathematical induction?
If the sum of the first n positive integers is given by the formula $S_n = \frac{n(n+1)}{2}$, which of the following represents the inductive step in proving this formula using mathematical induction?
Which of the following statements is the negation of the statement 'p ↔ q'?
Which of the following statements is the negation of the statement 'p ↔ q'?
Which of the following statements is a tautology?
Which of the following statements is a tautology?
What is the theorem proved in the text using proof by contradiction?
What is the theorem proved in the text using proof by contradiction?
In logical terms, what does ¬(p ∨ q) ≡ ¬p ∧ ¬q represent?
In logical terms, what does ¬(p ∨ q) ≡ ¬p ∧ ¬q represent?
If Lucas does not have a cellphone or a laptop computer, how would this be expressed using De Morgan's laws?
If Lucas does not have a cellphone or a laptop computer, how would this be expressed using De Morgan's laws?
What is the role of De Morgan's laws in logic?
What is the role of De Morgan's laws in logic?
In the context of proof by contradiction, what does it mean to have a contradiction?
In the context of proof by contradiction, what does it mean to have a contradiction?
How does mathematical induction differ from proof by contradiction?
How does mathematical induction differ from proof by contradiction?
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Study Notes
Mathematical Induction
- Key Step: The inductive step assumes the statement is true for an arbitrary integer k and then proves that it must also be true for the next integer k+1.
- Basis Step Purpose: Establishes the truth of the statement for the initial or smallest value of the integer (often n = 1).
- Conclusion Purpose: Concludes that the statement is true for all integers greater than or equal to the initial value.
Proof by Contradiction
- Logical Structure: Assumes the negation of the statement to be proven (¬p) and arrives at a contradiction (¬r).
- ¬p Role: Represents the assumption that the statement being proven is false.
- ¬r Role: Represents the contradiction reached, showing the initial assumption of ¬p must be false, implying the original statement is true.
Logical Equivalences
- Valid Logical Equivalences: p ∨ ¬p ≡ T (Law of Excluded Middle), ¬(p ∧ q) ≡ ¬p ∨ ¬q (De Morgan's Law)
- Contrapositive: ¬q → ¬p
- Negation of p ↔ q: p ⊕ q (exclusive or/XOR)
Tautology
- Tautology: p ∨ ¬p ≡ T (Law of Excluded Middle)
Proof by Contradiction Example
- Theorem: There are infinitely many prime numbers.
De Morgan's Laws
- ¬(p ∨ q) ≡ ¬p ∧ ¬q
- Lucas's Case: ¬(Lucas has a cellphone ∨ Lucas has a laptop) ≡ ¬(Lucas has a cellphone) ∧ ¬(Lucas has a laptop)
- De Morgan's Law Role: Demonstrate logical equivalences, simplifying complex logical statements by transforming negations of conjunctions and disjunctions.
Proof by Contradiction vs. Mathematical Induction
- Contradiction: Starts with an assumption and derives a contradiction to prove the statement.
- Mathematical Induction: Proves a statement is true for all integers by establishing a base case and an inductive step.
- Difference: Contradiction relies on creating a contradiction, while induction relies on establishing a pattern and extending it to all integers.
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