# Problem Set 2 >> Introduction to Mathematical Thinking

## Problem Set 2 >> Introduction to Mathematical Thinking

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#### Problem Set 2

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Question 1

Which of the following conditions are necessary for the natural number n to be divisible by 6? Select all those you believe are necessary. [6 points]

6 / 6 points

n is divisible by 3.

Correct

n is divisible by 9.

n is divisible by 12.

n=24.

n^2 is divisible by 3.

Correct

n is even and divisible by 3.

Correct

Question 2

Which of the following conditions are sufficient for the natural number n to be divisible by 6? Select all those you believe are sufficient. [6 points]

6 / 6 points

n is divisible by 3.

n is divisible by 9.

n is divisible by 12.

Correct

n=24.

Correct

n^2 is divisible by 3.

n is even and divisible by 3.

Correct

Question 3

Which of the following conditions are necessary and sufficient for the natural number n to be divisible by 6? Select all those you believe are necessary and sufficient. [6 points]

6 / 6 points

n is divisible by 3.

n is divisible by 9.

n is divisible by 12.

n=24.

n^2 is divisible by 3.

n is even and divisible by 3.

Correct

Question 4

Identify the antecedent in the conditional ”If the apples are red, they are ready to eat.” [1 point]

1 / 1 point

THE APPLES ARE RED

THE APPLES ARE READY TO EAT

Correct

Question 5

Identify the antecedent in the conditional ”The differentiability of a function f is sufficient for f to be continuous.” [1 point]

1 / 1 point

f IS DIFFERENTIABLE

f IS CONTINUOUS

Correct

Question 6

Identify the antecedent in the conditional ”A function f is bounded if f is integrable.” [1 point]

1 / 1 point

f IS BOUNDED

f IS INTEGRABLE

Correct

Question 7

Identify the antecedent in the conditional ”A sequence S is bounded whenever S is convergent.” [1 point]

1 / 1 point

S IS BOUNDED

S IS CONVERGENT

Correct

Question 8

Identify the antecedent in the conditional ”It is necessary that n is prime in order for 2^n – 1 to be prime.” [1 point]

1 / 1 point

n IS PRIME

2^n – 1 IS PRIME

Correct

Question 9

Identify the antecedent in the conditional ”The team wins only when Karl is playing.” [1 point]

1 / 1 point

THE TEAM WINS

KARL IS PLAYING

Correct

Question 10

Identify the antecedent in the conditional ”When Karl plays the team wins.” [1 point]

1 / 1 point

THE TEAM WINS

KARL PLAYS

Correct

Question 11

Identify the antecedent in the conditional ”The team wins when Karl plays.” [1 point]

1 / 1 point

THE TEAM WINS

KARL PLAYS

Correct

Question 12

For natural numbers m, n, is it true that mn is even iff m and n are even? [2 points]

2 / 2 points

Yes

No

Correct

Question 13

Is it true that mn is odd iff m and n are odd? [2 points]

2 / 2 points

Yes

No

Correct

Question 14

Which of the following pairs of propositions are equivalent? Select all you think are equivalent. [6 points]

6 / 6 points

\neg P \vee Q \ , \ P \Rightarrow Q

Correct

\neg (P \vee Q) \ , \ \neg P \wedge \neg Q

Correct

\neg P \vee \neg Q \ , \ \neg (P \vee \neg Q)

\neg (P \wedge Q) \ , \ \neg P \vee \neg Q

Correct

\neg (P \Rightarrow (Q \wedge R)) \ , \ \neg (P \Rightarrow Q) \vee \neg (P \Rightarrow R)

Correct

P \Rightarrow (Q \Rightarrow R) \ , \ (P \wedge Q) \Rightarrow R

Correct

Question 15

A major focus of this course is learning how to assess mathematical reasoning. How good you are at doing that lies on a sliding scale. Your task is to evaluate this purported proof according to the course rubric.

PS2_Q15.pdfPDF File

Enter your evaluation (which should be a whole number between 0 and 24, inclusive) in the box. An answer within 4 points of the instructor’s evaluation counts as correct. [5 points]

You should read the website section “Using the evaluation rubric” (and watch the associated short explanatory video) before attempting this question. There will be many more proof evaluation questions as the course progresses.

NOTE: The scoring system for proof evaluation questions is somewhat arbitrary, due to limitations of the platform. But the goal is to provide opportunities for you to reflect on what makes an argument a good proof, and you are allowed to repeat the Problem Sets as many times as it takes to be able to progress. Your “score” is simply feedback information. Moreover, the “passing grade” for Problem Sets is a low 35%.

5 / 5 points

18

Correct