Mathematical Induction and Proofs by Contradiction

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Questions and Answers

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?

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?

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?

To state that the statement is true for all integers n ≥ b Signup and view all the answers

In the proof by contradiction example, what is the role of the statement ¬p?

It is the negation of the original statement Signup and view all the answers

What is the role of the statement ¬r in the proof by contradiction example?

It is a statement that is used to derive a contradiction Signup and view all the answers

Which of the following is a valid logical equivalence?

(p → q) ∧ (p → r) ≡ p → (q ∧ r) Signup and view all the answers

Which of the following statements is the contrapositive of the statement 'p → q'?

¬q → ¬p Signup and view all the answers

In a proof by mathematical induction, what is the purpose of the basis step?

To show that the statement P(1) is true Signup and view all the answers

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?

Assume that $S_k = \frac{k(k+1)}{2}$ is true for some positive integer k, and show that $S_{k+1} = \frac{(k+1)((k+1)+1)}{2}$ is also true. Signup and view all the answers

Which of the following statements is the negation of the statement 'p ↔ q'?

¬p ↔ ¬q Signup and view all the answers

Which of the following statements is a tautology?

(p ∧ q) → (p ∨ q) Signup and view all the answers

What is the theorem proved in the text using proof by contradiction?

If $3n + 2$ is odd, then $n$ is odd. Signup and view all the answers

In logical terms, what does ¬(p ∨ q) ≡ ¬p ∧ ¬q represent?

Negation of conjunctions Signup and view all the answers

If Lucas does not have a cellphone or a laptop computer, how would this be expressed using De Morgan's laws?

Lucas has neither a cellphone nor a laptop computer. Signup and view all the answers

What is the role of De Morgan's laws in logic?

To simplify logical expressions Signup and view all the answers

In the context of proof by contradiction, what does it mean to have a contradiction?

The assumptions made are inconsistent. Signup and view all the answers

How does mathematical induction differ from proof by contradiction?

Mathematical induction involves proving a statement for all natural numbers. Signup and view all the answers

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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Description

Learn about mathematical induction and proofs by contradiction with exercises taken from the book of Rosen. Identify the conclusion of the inductive step and state the final conclusion after completing the basis step and the inductive step.