Solve the following exercise.
11. 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.
To solve this type of exercise, we don’t need extra information/definitions. We can just read the sentence and substitute each proposition for the corresponding propositional variable (p or q). Apart from that, we should consider if there is a negation involved.
That’s it. Let’s solve the exercises.
a) It is below freezing and snowing
Answer: p ^ q
b) It is below freezing but not snowing
Notice that in this case, q is negated. q is the proposition it is snowing. Here, we have to write “not snowing”.
Answer: p ^ ¬ q
c) It is not below freezing and it is not snowing
This case is similar to the previous one. Both p and q are negated.
Answer: ¬p ^ ¬q
d) It is either snowing or below freezing (or both).
From the clarification “or both”, we should infer that is not the exclusive or (xor or ⊕)
Answer: p v q
e) If it is below freezing, it is also snowing
Answer: p -> q
f) Either it is below freezing or it is snowing, but it is not snowing if it is below freezing
Notice that in this case, the way that “either” is used in the sentence means that is one or the other one, but not both. You can also see that in exercise d), “either” is also used but then there is a clarification that can be both. Because of this, we can conclude that in this case, we must use the exclusive or.
Answer: (p ⊕ q) ^ (p -> ¬q)
g) That it is below freezing is necessary and sufficient for it to be snowing
Notice that necessary and sufficient is translated to a biconditional statement.
Answer: p ⟷ q
Related posts:
- How to create the truth table for a compound proposition
- Propositional Logic: Exercise 6
- What is the negation of each of these propositions?
- What are propositional equivalences in Discrete Mathematics?
- Which of these sentences are propositions? What are the truth values of those that are propositions?
- For each of these sentences, state what the sentence means if the logical connective or is an inclusive or (that is, a disjunction) versus an exclusive or. Which of these meanings of or do you think is intended?
