Question

Write a conditional statement. Write the converse, inverse, and contrapositive for your statement and determine the truth value of each. If the statements truth value is false, give a counter example.

Answer 1

Conditional statement is a statement with a hypotesis and a conclusion:

$If \text{ \underline{ hypothesis } } p, then \text{ \underline { conclusion } } q$

or mathematically $p\rightarrow q$.

Converse statement of $p\rightarrow q$ is statement $q\rightarrow p$.

If you negate (that means stick a "not" in front of) both the hypothesis and conclusion, you get the inverse:

$\neg p\rightarrow \neg q$.

Finally, if you negate everything and flip p and q (taking the inverse of the converse) then you get the contrapositive:

$\neg q\rightarrow \neg p$.

Example:

1. Conditional statement: If I am sleeping, then I have closed eyes. (true)

2. Converse statement: If I have closed eyes, then I'm sleeping. (not necessarily true)

3. Inverse statement: If I'm not sleeping, then I haven't closed eyes. (not necessarily true)

4. Contrapositive statement: If I haven't closed eyes, then I'm not sleeping. (true)