18 Questions
What is the key step in the proof by mathematical induction?
Stating the inductive step
What is the purpose of the basis step in a mathematical induction proof?
To show the statement is true for the first value
What is the logical structure of a proof by contradiction?
Assume the statement is false, and derive a contradiction
What is the purpose of the conclusion in a mathematical induction proof?
To state that the statement is true for all integers n ≥ b
In the proof by contradiction example, what is the role of the statement ¬p?
It is the negation of the original statement
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
Which of the following is a valid logical equivalence?
(p → q) ∧ (p → r) ≡ p → (q ∧ r)
Which of the following statements is the contrapositive of the statement 'p → q'?
¬q → ¬p
In a proof by mathematical induction, what is the purpose of the basis step?
To show that the statement P(1) is true
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.
Which of the following statements is the negation of the statement 'p ↔ q'?
¬p ↔ ¬q
Which of the following statements is a tautology?
(p ∧ q) → (p ∨ q)
What is the theorem proved in the text using proof by contradiction?
If $3n + 2$ is odd, then $n$ is odd.
In logical terms, what does ¬(p ∨ q) ≡ ¬p ∧ ¬q represent?
Negation of conjunctions
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.
What is the role of De Morgan's laws in logic?
To simplify logical expressions
In the context of proof by contradiction, what does it mean to have a contradiction?
The assumptions made are inconsistent.
How does mathematical induction differ from proof by contradiction?
Mathematical induction involves proving a statement for all natural numbers.
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.
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