Problem Set 2 >> Introduction to Mathematical Thinking

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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

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.

Correct

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

$n$ is even and divisible by 3.

Correct

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

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

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

$n = 24$ .

Correct

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.

Correct

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

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

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

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

Well done. This one is tricky.

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 $m n$ is even iff $m$ and $n$ are even? [2 points]

2 / 2 points

Yes

Correct

Correct. $\ n=3$ provides a counterexample.

Question 13

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

2 / 2 points

Yes

Correct

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

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