Questions: Logic Symbolic translation involving three statements Consider statements p, q, and r. p: It is cloudy. q: Lisa is wearing a coat. r: It is fall. For each part below, fill in the symbolic form. Descriptive form Symbolic form (a) It is cloudy if and only if it is fall, and Lisa is not wearing a coat. (b) It is fall, or it is not cloudy and Lisa is wearing a coat.

Logic
Symbolic translation involving three statements

Consider statements p, q, and r.
p: It is cloudy.
q: Lisa is wearing a coat.
r: It is fall.
For each part below, fill in the symbolic form.
Descriptive form
Symbolic form
(a) It is cloudy if and only if it is fall, and Lisa is not wearing a coat.
(b) It is fall, or it is not cloudy and Lisa is wearing a coat.

Transcript text: Logic Symbolic translation involving three statements Consider statements $p, q$, and $r$. $p:$ It is cloudy. $q$ : Lisa is wearing a coat. $r$ : It is fall. For each part below, fill in the symbolic form. Descriptive form Symbolic form (a) It is cloudy if and only if it is fall, and Lisa is not wearing a coat. (b) It is fall, or it is not cloudy and Lisa is wearing a coat.

Solution

Solution Steps

Solution Approach

(a) To translate the descriptive form into symbolic form, we need to use logical connectives. "It is cloudy if and only if it is fall" translates to $ p \leftrightarrow r $. "Lisa is not wearing a coat" translates to $ \neg q $. Combining these, we get $ (p \leftrightarrow r) \land \neg q $.

(b) "It is fall" translates to $ r $. "It is not cloudy" translates to $ \neg p $. "Lisa is wearing a coat" translates to $ q $. Combining these, we get $ r \lor (\neg p \land q) $.

Step 1: Define the Statements

We are given three statements:

$ p $: It is cloudy.
$ q $: Lisa is wearing a coat.
$ r $: It is fall.

Step 2: Translate Descriptive Form to Symbolic Form

(a) The descriptive form "It is cloudy if and only if it is fall, and Lisa is not wearing a coat" translates to: \[ (p \leftrightarrow r) \land \neg q \]

(b) The descriptive form "It is fall, or it is not cloudy and Lisa is wearing a coat" translates to: \[ r \lor (\neg p \land q) \]

Step 3: Simplify and Verify the Symbolic Forms

Using the Python output, we verify the symbolic forms:

For (a): $ \neg q \land (p \leftrightarrow r) $
For (b): $ r \lor (q \land \neg p) $

Final Answer

(a) The symbolic form of "It is cloudy if and only if it is fall, and Lisa is not wearing a coat" is: \[ \boxed{\neg q \land (p \leftrightarrow r)} \]

(b) The symbolic form of "It is fall, or it is not cloudy and Lisa is wearing a coat" is: \[ \boxed{r \lor (q \land \neg p)} \]

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