Question

Write a conditional statement. Write the converse, inverse and contrapositive for your statement and determine the truth value of each. if a statement's truth value is false, give a conunterexample.

Answer 1

if the weather is nice, then i will go out

converse q---->p if i go out, then the weather is nice

inverse ~p---.~q if the weather is not nice, then i will not go out

contrapositive ~q--->~p if i don't go out, then the weather is not nice

Explanation:

consider the statement below

if the weather is nice, then i will go out

we designate the first part of the statement as p, and the other part as q

p=if the weather is nice

q=then i will go out

we can get the negations for this statement as

~p=if the weather is not nice

~q=then i will not go out

converse q---->p if i go out, then the weather is nice

inverse ~p---.~q if the weather is not nice, then i will not go out

contrapositive ~q--->~p if i don't go out, then the weather is not nice

Answer 2

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:

`\\eg p\rightarrow \\eg q`.

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

`\\eg q\rightarrow \\eg p`.

Example:

1. Statement: If it is raining, then I'm at home. (true)

2. Converse: If I'm at home, then it is raining. (not necessarily true)

3. Inverse: If it is not raining, then I'm not at home. (not necessarily true)

4. Contrapositive: If I'm not at home, then it is not raining. (true)