Question

Let p and q be the propositions p : It is below freezing. q : It is snowing. Write these propositions using p and q and logical connectives (including negations).

a) It is below freezing and snowing.
b) It is below freezing but not snowing.
c) It is not below freezing and it is not snowing.
d) It is either snowing or below freezing (or both).
e) If it is below freezing, it is also snowing.
f )Either it is below freezing or it is snowing, but it is not snowing if it is below freezing.
g) That it is below freezing is necessary and sufficient for it to be snowing.

Short answer

1. $\mathrm{p}\wedge \mathrm{q}$
2. $\mathrm{p}\wedge \neg \mathrm{q}$
3. $\neg \mathrm{p}\wedge \neg \mathrm{q}$
4. $\mathrm{p}\vee \mathrm{q}$
5. $\mathrm{p}\to \mathrm{q}$
6. $(\mathrm{p}\vee \mathrm{q})\wedge (\mathrm{p}\to \neg \mathrm{q})$
7. $\mathrm{p}\leftrightarrow \mathrm{q}$
   

Step-by-step solution

Definition of proposition

Proposition:
1. It is a declarative statement which can be either true or false.
2. It cannot be both true and false simultaneously.

Proposition using p and q for a)

p: It is below freezing.

q: It is snowing.

It is below freezing and snowing.

The above given statement means $\mathrm{p}\wedge \mathrm{q}$.

Step 3:Proposition using p and q for b)

It is below freezing but not snowing.

The above given statement means $\mathrm{p}\wedge \neg \mathrm{q}$.

Proposition using p and q for c)

It is not below freezing and it is not snowing.

The above given statement means $\neg \mathrm{p}\wedge \neg \mathrm{q}$.

Proposition using p and q for d)

It is either snowing or below freezing (or both).

Then the given statement means $\mathrm{p}\vee \mathrm{q}$.

Proposition using p and q for e)

If it is below freezing, it is also snowing.

Then the given statement means$\mathrm{p}\to \mathrm{q}$.

Proposition using p and q for f)

Either it is below freezing or it is snowing, but it is not snowing if it is below freezing.

The given statement means $(\mathrm{p}\vee \mathrm{q})\wedge (\mathrm{p}\to \neg \mathrm{q})$.

Proposition using p and q for g)

That it is below freezing is necessary and sufficient for it to be snowing.

The given statement means $\mathrm{p}\leftrightarrow \mathrm{q}$.