Chapter 2 Flashcards
Which of the following is the correct method to disprove a universal statement?
A. Show a specific case where the statement is true.
B. Show a specific case where the statement is false.
C. Prove the contrapositive of the statement.
D. Assume the opposite and reach a contradiction.
D. Assume the opposite and reach a contradiction.
Which of the following disproves the statement “All prime numbers are odd”?
A. Prime numbers cannot be odd.
B. The number 2 is a prime number and is even.
C. Odd numbers are always prime.
D. All numbers are divisible by 2.
B. The number 2 is a prime number and is even.
What is an exhaustive proof?
A. A proof that examines all possible cases to confirm a statement.
B. A proof that involves mathematical induction.
C. A proof that derives contradictions from assumptions.
D. A proof that assumes the negation of the statement.
A. A proof that examines all possible cases to confirm a statement.
Which of the following is best solved using an exhaustive proof?
A. “For all n, if n>1, then n^2>n.”
B. “The sum of two even numbers is always even.”
C. “Every number less than 10 is either prime or composite.”
D. “There exists an even number greater than 2.”
C. “Every number less than 10 is either prime or composite.”
In a direct proof, what is the main approach?
A. Start with the conclusion and derive the hypothesis.
B. Start with the hypothesis and logically deduce the conclusion.
C. Assume the conclusion is false and derive a contradiction.
D. Show a counterexample.
B. Start with the hypothesis and logically deduce the conclusion.
What is the contrapositive of the statement “If P, then Q”?
A. If Q, then P.
B. If not P, then not Q.
C. If not Q, then not P.
D. If P and Q, then true.
C. If not Q, then not P.
What is the key idea in proof by contradiction?
A. Prove the statement directly.
B. Assume the negation of the statement and derive a contradiction.
C. Assume the contrapositive and derive a conclusion.
D. Test all possible cases.
B. Assume the negation of the statement and derive a contradiction.
What are the steps involved in the first principle of induction?
A. Prove the base case, prove the induction hypothesis, and prove the inductive step.
B. Prove the base case and assume the induction hypothesis is true.
C. Prove the base case and prove the contrapositive.
D. Assume the base case and derive a contradiction.
A. Prove the base case, prove the induction hypothesis, and prove the inductive step.
In a proof by induction, what is the role of the inductive step?
A. Verify the statement for the smallest value of n.
B. Assume the statement is true for n=k and prove it for n=k+1.
C. Prove the statement for all possible values of n.
D. Show the statement is false for at least one n.
B. Assume the statement is true for n=k and prove it for n=k+1.
Which of the following statements can be proved using induction?
A. “Every even number greater than 2 is the sum of two primes.”
B. “The sum of the first n odd numbers is n^2.”
C. “There exists an even prime number.”
D. “The product of two odd numbers is odd.”
B. “The sum of the first n odd numbers is n^2.”
What is the advantage of using proof by contraposition?
A. It always requires fewer steps than a direct proof.
B. It allows you to prove a statement by focusing on its equivalent contrapositive.
C. It avoids proving the base case in an induction proof.
D. It provides counterexamples to disprove the original statement.
B. It allows you to prove a statement by focusing on its equivalent contrapositive.
