# Problem Set 2 >> Introduction to Mathematical Thinking

## Problem Set 2 >> Introduction to Mathematical Thinking

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#### Problem Set 2

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]

$n$ is divisible by 3.

You have to select conditions that follow from $n$ being divisible by 6.

$n$ is divisible by 9.

$n$ is divisible by 12.

$n=24$.

$n_{2}$ is divisible by 3.

You have to select conditions that follow from $n$ being divisible by 6.

$n$ is even and divisible by 3.

You have to select conditions that follow from $n$ being divisible by 6.

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]

$n$ is divisible by 3.

$n$ is divisible by 9.

$n$ is divisible by 12.

You have to select conditions that imply $n$ is divisible by 6.

$n=24$.

You have to select conditions that imply $n$ is divisible by 6.

$n_{2}$ is divisible by 3.

$n$ is even and divisible by 3.

You have to select conditions that imply $n$ is divisible by 6.

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]

$n$ 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.

You have to select conditions that both imply and are a consequence of $n$ being divisible by 6.

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

THE APPLES ARE RED

THE APPLES ARE READY TO EAT

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

$f$ IS DIFFERENTIABLE

$f$ IS CONTINUOUS

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

$f$ IS BOUNDED

$f$ IS INTEGRABLE

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

S IS BOUNDED

S IS CONVERGENT

Identify the antecedent in the conditional ”It is necessary that $n$ is prime in order for $_{n}−1$ to be prime.” [1 point]

$n$ IS PRIME

$_{n}−1$ IS PRIME

Well done. This one is tricky.

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

THE TEAM WINS

KARL IS PLAYING

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

THE TEAM WINS

KARL PLAYS

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

THE TEAM WINS

KARL PLAYS

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

Yes

No

Correct. $m=2,n=3$ provides a counterexample.

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

Yes

No

Correct. The question splits into two parts: (i) does $mn$ being odd imply both $m$ and $n$ are odd, and (ii) does $m$ and $n$ being odd imply $mn$ is odd. The answer in both cases is Yes.

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

$¬P∨Q,P⇒Q$

Yes, these are equivalent. If you evaluate the two truth tables correctly, you will find they have the same final columns.

$¬(P∨Q),¬P∧¬Q$

Yes, these are equivalent. If you evaluate the two truth tables correctly, you will find they have the same final columns.

$¬P∨¬Q,¬(P∨¬Q)$

$¬(P∧Q),¬P∨¬Q$

Yes, these are equivalent. If you evaluate the two truth tables correctly, you will find they have the same final columns.

$¬(P⇒(Q∧R)),¬(P⇒Q)∨¬(P⇒R)$

$P⇒(Q⇒R),(P∧Q)⇒R$

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.

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%.

Good grade. The proof of the left-to-right implication is correct and well laid out, so you should end up giving at least 12 out of a possible 24. The author is wrong to assume the other implication is valid (in fact the two are not equivalent), but the logical structure and clarity is good, so 18 would be a fair grade. WATCH THE TUTORIAL VIDEO.