I need help on this question.
Write the contrapositive,converse,and inverse of the statement.Determine if its true or false.If its false give a coynterexample.
If I live in Redlands,then I live in California.
To write the contrapositive, converse, and inverse of the given statement, "If I live in Redlands, then I live in California," we will first need to identify the hypothesis (p) and the conclusion (q) in the statement.
The hypothesis is "I live in Redlands," and the conclusion is "I live in California."
1. The contrapositive is formed by switching the hypothesis and conclusion and negating both:
- Contrapositive: "If I do not live in California, then I do not live in Redlands."
2. The converse is formed by switching the hypothesis and conclusion:
- Converse: "If I live in California, then I live in Redlands."
3. The inverse is formed by negating both the hypothesis and conclusion:
- Inverse: "If I do not live in Redlands, then I do not live in California."
Now let's determine if each of these statements is true or false:
- The original statement is: "If I live in Redlands, then I live in California."
- This statement is true because if you live in Redlands, it is guaranteed that you live in California.
- The contrapositive is: "If I do not live in California, then I do not live in Redlands."
- This statement is also true. If you do not live in California, it means you cannot live in Redlands because Redlands is located in California.
- The converse is: "If I live in California, then I live in Redlands."
- This statement is false because it is possible to live in California without living specifically in the city of Redlands. Therefore, the converse requires a counterexample.
- The inverse is: "If I do not live in Redlands, then I do not live in California."
- This statement is also false because it is possible to live in California without living specifically in Redlands. Again, a counterexample is needed.
To provide a counterexample for the converse and inverse, we can say:
Counterexample for the converse: "Living in California does not necessarily mean living in Redlands. For example, I live in Los Angeles, California, but not in Redlands."
Counterexample for the inverse: "Not living in Redlands does not necessarily mean not living in California. For example, I live in San Diego, California, but not in Redlands."