Question
Consider statements \( p, q \), and \( r \). \[ p \text { : It is cloudy. } \] \( q: \) Mary is wearing a coat. \( r: \) It is fall. For each part below, fill in the symbolic form. (a) Mary is wearing a coat, or if it is not fall then it is not cloudy. (b) If Mary is not wearing a coat then it is cloudy, if and only if it is fall.
Ask by Salinas Sullivan. in the United States
Feb 28,2025
Upstudy AI Solution
Tutor-Verified Answer
Answer
(a) q ∨ (¬r → ¬p)
(b) (¬q → p) ↔ r
Solution
We are given the following symbols:
p: It is cloudy.
q: Mary is wearing a coat.
r: It is fall.
(a) "Mary is wearing a coat, or if it is not fall then it is not cloudy."
Symbolically: q ∨ (¬r → ¬p)
(b) "If Mary is not wearing a coat then it is cloudy, if and only if it is fall."
Symbolically: (¬q → p) ↔ r
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Explain 
Simplify this solution Extra Insights
(a) The symbolic form is \( q \lor (\neg r \rightarrow \neg p) \). This translates to "Mary is wearing a coat, or if it is not fall then it is not cloudy," capturing the logical relationship between the variables as described. (b) The symbolic form is \( \neg q \rightarrow (p \leftrightarrow r) \). This represents "If Mary is not wearing a coat, then it is cloudy if and only if it is fall," linking the coat wearing status to the cloudiness and the season.
