It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a cano... It is not sunny this afternoon and it is colder than yesterday. We will go swimming only if it is sunny. If we do not go swimming, then we will take a canoe trip. If we take a canoe trip, then we will be home by sunset. Compute a logical expression using propositional variables (P, Q, R, S, and T) and logical connectives for the above statements. Read more

Steps to Solve

1. Define Propositional Variables
   Assign meanings to the propositional variables:

• Let $P$: It is sunny this afternoon.
• Let $Q$: We will go swimming.
• Let $R$: We will take a canoe trip.
• Let $S$: We will be home by sunset.
• Let $T$: It is colder than yesterday.

2. Translate Statements to Logical Expressions
   Translate the statements into logical expressions using the variables defined:

• "It is not sunny this afternoon and it is colder than yesterday" translates to:
  $$ \neg P \land T $$

• "We will go swimming only if it is sunny" translates to:
  $$ Q \rightarrow P $$

• "If we do not go swimming, we will take a canoe trip" translates to:
  $$ \neg Q \rightarrow R $$

• "If we take a canoe trip, then we will be home by sunset" translates to:
  $$ R \rightarrow S $$

3. Combine the Logical Expressions
   Combine all the translated expressions using logical connectives:

• The entire expression is:
  $$ (\neg P \land T) \land (Q \rightarrow P) \land (\neg Q \rightarrow R) \land (R \rightarrow S) $$

4. Final Expression
   The final logical expression representing all the statements is:
   $$ \neg P \land T \land (Q \rightarrow P) \land (\neg Q \rightarrow R) \land (R \rightarrow S) $$