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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

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(a) q ∨ (¬r → ¬p) (b) (¬q → p) ↔ r

Solución

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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(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.

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