Question

Given the following sentences, convert them into an appropriate propositional

algebra form, assuming R represents the statement "It is raining" and S represents
the statement "It is snowing".

1. It is raining.
2. It is not raining.
3. It is raining but not snowing.
4. If it is not raining, then it is snowing.
5. It is not the case that if it is snowing then it is raining.
6. It is neither raining nor snowing.
7. If it is both raining and snowing, then it is snowing.
8. Either it is both raining and snowing or it is snowing but not raining.
Build truth tables for problems 2,4,6, and 8.

Answer 1

Here is the propositional algebra form for the given sentences. Truth tables are also provided for problems 2, 4, 6, and 8.

Step-by-step explanation:

1. R: It is raining
2. ~R: It is not raining
3. R ∧ ~S: It is raining but not snowing
4. ~R ⟶ S: If it is not raining, then it is snowing
5. ~(S ⟶ R): It is not the case that if it is snowing then it is raining
6. ~R ∧ ~S: It is neither raining nor snowing
7. (R ∧ S) ⟶ S: If it is both raining and snowing, then it is snowing
8. (R ∧ S) ∨ (S ∧ ~R): Either it is both raining and snowing or it is snowing but not raining

Truth table for problem 2:

R ~R

T F

Truth table for problem 4:

R S ~R ⟶ S

T T T

T F F

F T T

F F T

Truth table for problem 6:

R S ~R ∧ ~S

T T F

T F F

F T F

F F T

Truth table for problem 8:

R S (R ∧ S) ∨ (S ∧ ~R)

T T T

T F F

F T T

F F T