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

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. 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)

Answer 2

Answer:

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

Step-by-step 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