Let p -> q represent the conditional statement All hummingbirds love nectar.
This is equivalent to p = If it is a hummingbird and q = then it loves nectar.
State your answers to the questions below both in words and in symbols.
1. What is the contrapositive of this statement?
2. What is the converse of this statement?
3. What is the inverse of this statement?
4. What is the negation of this statement?
Can you give an answer to the same four questions if the original statement is "If it freezes in Sept., then the tomato plants stop producing. For this second example, let r -> s represent this conditional statement.
5. What is the contrapositive of this statement?
6. What is the converse of this statement?
7. What is the inverse of this statement?
8. What is the negation of this statement?
To answer these questions, we need to understand the various forms of statements:
1. Contrapositive: To find the contrapositive of a conditional statement, we switch and negate both the hypothesis (p) and the conclusion (q) of the original statement.
- Original statement: p -> q ("All hummingbirds love nectar")
- Contrapositive: ~q -> ~p ("If it does not love nectar, then it is not a hummingbird")
2. Converse: The converse of a conditional statement involves switching the hypothesis and the conclusion while keeping the same directionality.
- Original statement: p -> q ("All hummingbirds love nectar")
- Converse: q -> p ("If it loves nectar, then it is a hummingbird")
3. Inverse: The inverse of a conditional statement entails negating both the hypothesis (p) and the conclusion (q) of the original statement.
- Original statement: p -> q ("All hummingbirds love nectar")
- Inverse: ~p -> ~q ("If it is not a hummingbird, then it does not love nectar")
4. Negation: The negation of a conditional statement negates the entire statement.
- Original statement: p -> q ("All hummingbirds love nectar")
- Negation: ~(p -> q) ("It is not the case that all hummingbirds love nectar")
For the second example: "If it freezes in Sept., then the tomato plants stop producing":
5. Contrapositive: ~s -> ~r ("If the tomato plants do not stop producing, then it did not freeze in Sept.")
6. Converse: s -> r ("If the tomato plants stop producing, then it froze in Sept.")
7. Inverse: ~r -> ~s ("If it did not freeze in Sept., then the tomato plants do not stop producing.")
8. Negation: ~(r -> s) ("It is not the case that if it freezes in Sept., then the tomato plants stop producing.")
Remember, when dealing with conditional statements, it is essential to properly manipulate the hypotheses and conclusions to find the contrapositive, converse, inverse, and negation.