# Problem Set 1 >> Introduction to Mathematical Thinking

## Problem Set 1 >> Introduction to Mathematical Thinking

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#### Problem Set 1

Is it possible for one of $(ϕ∧ψ)∧θ$ and $ϕ∧(ψ∧θ)$ to be true and the other false? (If not, then the associative property holds for conjunction.) [Score: 5 points]

Yes

No

Correct!

Is it possible for one of $(ϕ∨ψ)∨θ$ and $ϕ∨(ψ∨θ)$ to be true and the other false? (If not, then the associative property holds for disjunction.) [Score: 5 points]

Yes

No

Correct!

Is it possible for one of $ϕ∧(ψ∨θ)$ and $(ϕ∧ψ)∨(ϕ∧θ)$ to be true and the other false? (If not, then the distributive property holds for conjunction across disjunction.) [Score: 5 points]

Yes

No

Correct!

Is it possible for one of $ϕ∨(ψ∧θ)$ and $(ϕ∨ψ)∧(ϕ∨θ)$ to be true and the other false? (If not, then the distributive property holds for disjunction across conjunction.) [Score: 5 points]

Yes

No

Correct!

Is showing that the negation $¬ϕ$ is true equivalent to showing that $ϕ$ is false? [Score: 5 points]

Yes

No

Correct.

Assuming you know nothing more about Alice, which of (a) – (e) is most likely? (Or does (f) hold?) [Score: 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! Conjoining any second requirement makes it less likely to be true.

Assuming you know nothing more about Alice, which of (a) – (e) is most likely? (Or does (f) hold?) [Score: 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! Disjoining a second requirement makes it more likely to be true.

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]

$(x>0)∧(x≤10)$ means $0≤x≤10$

$(x≥0)∧(x_{2}<9)$ means $0≤x<3$

This one is true.

$(x≥0)∧(x≤0)$ means $x=0$

This one is true.

There is no $x$ for which $(x<4)∧(x>4)$

This one is true.

$−5≤x≤5$ means $x$ is at most 5 units from 0.

This one is true.

$−5<x<5$ implies that $x$ cannot be exactly 5 units from 0.

This one is true.

$(x≥0)∨(x<0)$

This one is true.

$(0=1)∨(x_{2}≥0)$

This one is true.

If $(x>0∨x<0)$ then $x 0$.

This one is true.

If $x_{2}=9$ then $(x=3∨x=−3)$.

This one is true.