To determine the truth value of each statement given the conditions, let's first understand the logical relationships and then directly derive the true statements based on the results provided.
Original Conditional Statement [tex]\( p \rightarrow q \)[/tex]:
- "If we double the dimensions of a rectangle, then the area increases by a factor of 4."
- This statement is given to be true.
Inverse [tex]\( \sim p \rightarrow \sim q \)[/tex]:
- "If we do not double the dimensions of a rectangle, then the area does not increase by a factor of 4."
- This is not necessarily true. The rectangle could change dimensions in some other way that still results in the area increasing by a factor of 4.
Converse [tex]\( q \rightarrow p \)[/tex]:
- "If the area increases by a factor of 4, then we have doubled the dimensions of the rectangle."
- This assumes that increasing the area by a factor of 4 can only be achieved by doubling the dimensions. There might be other ways to achieve this area increase, so this is not necessarily true.
Contrapositive [tex]\( \sim q \rightarrow \sim p \)[/tex]:
- "If the area does not increase by a factor of 4, then we have not doubled the dimensions of the rectangle."
- This is logically equivalent to the original conditional statement [tex]\( p \rightarrow q \)[/tex], and thus true.
Based on these logical relationships and the provided numerical result:
1. [tex]\( p \rightarrow q \)[/tex] (If we double the dimensions, the area increases by a factor of 4. True as given)
2. [tex]\( \sim q \rightarrow \sim p \)[/tex] (If the area does not increase by a factor of 4, we did not double the dimensions. True as it is the contrapositive of the original conditional statement)
Therefore, the correct options are indeed:
1. [tex]\( p \rightarrow q \)[/tex]
2. [tex]\( \sim q \rightarrow \sim p \)[/tex]
Thus, the true options correspond to:
First option: [tex]\( p \rightarrow q \)[/tex]
Second option: [tex]\( \sim q \rightarrow \sim p \)[/tex]
These correspond to options 1 and 4 in the given multiple-choice question.
