# Problem Set 1 >> Introduction to Mathematical Thinking

## Problem Set 1 >> Introduction to Mathematical Thinking

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#### Problem Set 1

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Question 1

Is it possible for one of (\phi \wedge \psi) \wedge \theta and \phi \wedge (\psi \wedge \theta) to be true and the other false? (If not, then the associative property holds for conjunction.) [Score: 5 points]

5 / 5 points

Yes

No

Correct

Question 2

Is it possible for one of (\phi \vee \psi) \vee \theta and \phi \vee (\psi \vee \theta) to be true and the other false? (If not, then the associative property holds for disjunction.) [Score: 5 points]

5 / 5 points

Yes

No

Correct

Question 3

Is it possible for one of \phi \wedge (\psi \vee \theta) and (\phi \wedge \psi ) \vee (\phi \wedge \theta) to be true and the other false? (If not, then the distributive property holds for conjunction across disjunction.) [Score: 5 points]

5 / 5 points

Yes

No

Correct

Question 4

Is it possible for one of \phi \vee (\psi \wedge \theta) and (\phi \vee \psi ) \wedge (\phi \vee \theta) to be true and the other false? (If not, then the distributive property holds for disjunction across conjunction.) [Score: 5 points]

5 / 5 points

Yes

No

Correct

Question 5

Is showing that the negation \neg \phi is true equivalent to showing that \phi is false? [Score: 5 points]

5 / 5 points

Yes

No

Correct

Question 6

Assuming you know nothing more about Alice, which of (a) – (e) is most likely? (Or does (f) hold?) [Score: 5 points]

5 / 5 points

(a) Alice is a rock star and works in a bank.

(b) Alice is quiet and works in a bank.

(c) Alice is quiet and reserved and works in a bank.

(d) Alice is honest and works in a bank.

(e) Alice works in a bank.

(f) None of the above is more or less likely.

Correct

Question 7

Assuming you know nothing more about Alice, which of (a) – (e) is most likely? (Or does (f) hold?) [Score: 5 points]

5 / 5 points

(a) Alice is a rock star or she works in a bank.

(b) Alice is quiet and works in a bank.

(c) Alice is a rock star.

(d) Alice is honest and works in a bank.

(e) Alice works in a bank.

(f) None of the above is more or less likely.

Correct

Question 8

Identify which of the following are true (where x denotes an arbitrary real number). If you do not select a particular statement, the system will assume you think it is false. [Score: 5 points]

5 / 5 points

(x\gt 0) \wedge (x \leq 10) means 0 \leq x \leq 10

(x \geq 0) \wedge (x^2 \lt 9) means 0 \leq x \lt 3

Correct

(x \geq 0) \wedge (x \leq 0) means x=0

Correct

There is no x for which (x \lt 4) \wedge (x \gt 4)

Correct

– 5 \leq x \leq 5 means x is at most 5 units from 0.

Correct

-5 \lt x \lt 5 implies that x cannot be exactly 5 units from 0.

Correct

(x \geq 0) \vee (x \lt 0)

Correct

(0 = 1) \vee (x^2 \geq 0)

Correct

If (x \gt 0 \vee x \lt 0) then x \neq 0.

Correct

If x^2 = 9 then (x = 3 \vee x = -3).

Correct