Answer :
The contrapositive of a conditional statement is a concept in logic that is deeply intertwined with the conditional statement itself. To understand the contrapositive, let us first define a few terms and review the necessary steps.
### Step-by-Step Solution:
1. Define the Conditional Statement:
A conditional statement, represented as [tex]\( p \rightarrow q \)[/tex], reads as "if [tex]\( p \)[/tex] then [tex]\( q \)[/tex]." This means that if the condition [tex]\( p \)[/tex] is true, then the result [tex]\( q \)[/tex] must also be true.
2. Negations:
The symbol [tex]\( \sim \)[/tex] represents negation in logic. Thus:
- [tex]\( \sim p \)[/tex] means "not [tex]\( p \)[/tex]"
- [tex]\( \sim q \)[/tex] means "not [tex]\( q \)[/tex]"
3. Contrapositive Definition:
The contrapositive of the statement [tex]\( p \rightarrow q \)[/tex] is formed by negating both the hypothesis and the conclusion of the original statement and then reversing them. Therefore, the contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
4. Options Provided:
Examine each of the given options to determine which represents the contrapositive:
- [tex]\( q \rightarrow p \)[/tex]: This is simply the converse of the original statement, not the contrapositive.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]: This is the correct representation of the contrapositive.
- [tex]\( p \rightarrow q \)[/tex]: This is the original statement itself, not the contrapositive.
- [tex]\( -p \rightarrow -q \)[/tex]: The symbols [tex]\( -p \)[/tex] and [tex]\( -q \)[/tex] are unconventional but would imply the negation. However, this representation is incorrect because it does not reverse the order.
5. Select the Correct Answer:
Therefore, the correct answer which represents the contrapositive of [tex]\( p \rightarrow q \)[/tex] is:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
According to the options provided, this corresponds to:
[tex]\[ \boxed{2} \][/tex]
### Step-by-Step Solution:
1. Define the Conditional Statement:
A conditional statement, represented as [tex]\( p \rightarrow q \)[/tex], reads as "if [tex]\( p \)[/tex] then [tex]\( q \)[/tex]." This means that if the condition [tex]\( p \)[/tex] is true, then the result [tex]\( q \)[/tex] must also be true.
2. Negations:
The symbol [tex]\( \sim \)[/tex] represents negation in logic. Thus:
- [tex]\( \sim p \)[/tex] means "not [tex]\( p \)[/tex]"
- [tex]\( \sim q \)[/tex] means "not [tex]\( q \)[/tex]"
3. Contrapositive Definition:
The contrapositive of the statement [tex]\( p \rightarrow q \)[/tex] is formed by negating both the hypothesis and the conclusion of the original statement and then reversing them. Therefore, the contrapositive of [tex]\( p \rightarrow q \)[/tex] is [tex]\( \sim q \rightarrow \sim p \)[/tex].
4. Options Provided:
Examine each of the given options to determine which represents the contrapositive:
- [tex]\( q \rightarrow p \)[/tex]: This is simply the converse of the original statement, not the contrapositive.
- [tex]\( \sim q \rightarrow \sim p \)[/tex]: This is the correct representation of the contrapositive.
- [tex]\( p \rightarrow q \)[/tex]: This is the original statement itself, not the contrapositive.
- [tex]\( -p \rightarrow -q \)[/tex]: The symbols [tex]\( -p \)[/tex] and [tex]\( -q \)[/tex] are unconventional but would imply the negation. However, this representation is incorrect because it does not reverse the order.
5. Select the Correct Answer:
Therefore, the correct answer which represents the contrapositive of [tex]\( p \rightarrow q \)[/tex] is:
[tex]\[ \sim q \rightarrow \sim p \][/tex]
According to the options provided, this corresponds to:
[tex]\[ \boxed{2} \][/tex]
