Answer :
Let's analyze the logical statements provided:
1. [tex]\( p \leftrightarrow q \)[/tex]: This means "He is wearing a coat if and only if the temperature is below 30°F." In other words, [tex]\( p \leftrightarrow q \)[/tex] asserts that the presence of one condition necessarily implies the presence of the other, and vice versa.
2. [tex]\( p \wedge q \)[/tex]: This means "He is wearing a coat and the temperature is below 30°F". This is a conjunction which combines both statements [tex]\( p \)[/tex] and [tex]\( q \)[/tex] into a single statement that is true only if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true simultaneously.
Now let's examine each of the options provided in the question to see which one corresponds to [tex]\( p \wedge q \)[/tex]:
- Option 1: "He is wearing a coat or the temperature is below 30°F".
- This corresponds to [tex]\( p \vee q \)[/tex], which represents a logical "or" and is true if either [tex]\( p \)[/tex] or [tex]\( q \)[/tex] is true or both are true. This does not match our requirement of [tex]\( p \wedge q \)[/tex].
- Option 2: "He is wearing a coat and the temperature is below 30°F".
- This directly corresponds to [tex]\( p \wedge q \)[/tex]. It explicitly states that both conditions must be true at the same time.
- Option 3: "If he is wearing a coat, then the temperature is below 30°F".
- This corresponds to [tex]\( p \rightarrow q \)[/tex], which represents a logical "if-then" statement. This does not match our requirement of [tex]\( p \wedge q \)[/tex].
- Option 4: "If he is not wearing a coat, then the temperature is not below 30°F".
- This corresponds to [tex]\( \neg p \rightarrow \neg q \)[/tex], which is the contrapositive of [tex]\( p \rightarrow q \)[/tex]. This does not match our requirement of [tex]\( p \wedge q \)[/tex].
Based on the detailed analysis above, the correct representation of [tex]\( p \wedge q \)[/tex] is:
He is wearing a coat and the temperature is below 30°F.
So, the answer is:
Option 2: He is wearing a coat and the temperature is below 30°F.
1. [tex]\( p \leftrightarrow q \)[/tex]: This means "He is wearing a coat if and only if the temperature is below 30°F." In other words, [tex]\( p \leftrightarrow q \)[/tex] asserts that the presence of one condition necessarily implies the presence of the other, and vice versa.
2. [tex]\( p \wedge q \)[/tex]: This means "He is wearing a coat and the temperature is below 30°F". This is a conjunction which combines both statements [tex]\( p \)[/tex] and [tex]\( q \)[/tex] into a single statement that is true only if both [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are true simultaneously.
Now let's examine each of the options provided in the question to see which one corresponds to [tex]\( p \wedge q \)[/tex]:
- Option 1: "He is wearing a coat or the temperature is below 30°F".
- This corresponds to [tex]\( p \vee q \)[/tex], which represents a logical "or" and is true if either [tex]\( p \)[/tex] or [tex]\( q \)[/tex] is true or both are true. This does not match our requirement of [tex]\( p \wedge q \)[/tex].
- Option 2: "He is wearing a coat and the temperature is below 30°F".
- This directly corresponds to [tex]\( p \wedge q \)[/tex]. It explicitly states that both conditions must be true at the same time.
- Option 3: "If he is wearing a coat, then the temperature is below 30°F".
- This corresponds to [tex]\( p \rightarrow q \)[/tex], which represents a logical "if-then" statement. This does not match our requirement of [tex]\( p \wedge q \)[/tex].
- Option 4: "If he is not wearing a coat, then the temperature is not below 30°F".
- This corresponds to [tex]\( \neg p \rightarrow \neg q \)[/tex], which is the contrapositive of [tex]\( p \rightarrow q \)[/tex]. This does not match our requirement of [tex]\( p \wedge q \)[/tex].
Based on the detailed analysis above, the correct representation of [tex]\( p \wedge q \)[/tex] is:
He is wearing a coat and the temperature is below 30°F.
So, the answer is:
Option 2: He is wearing a coat and the temperature is below 30°F.
