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Let [tex]\( p \leftrightarrow q \)[/tex] represent the statement "He is wearing a coat if and only if the temperature is below [tex]\( 30^{\circ} F \)[/tex]."

Which represents [tex]\( p \wedge q \)[/tex]?

A. He is wearing a coat or the temperature is below [tex]\( 30^{\circ} F \)[/tex].

B. He is wearing a coat and the temperature is below [tex]\( 30^{\circ} F \)[/tex].

C. If he is wearing a coat, then the temperature is below [tex]\( 30^{\circ} F \)[/tex].

D. If he is not wearing a coat, then the temperature is not below [tex]\( 30^{\circ} F \)[/tex].

Sagot :

GinnyAnswer
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.
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