Question

Question #1) Write the inverse, converse, and contrapositive of the following conditional statement. Determine the truth value.

Answer 1

Answers:

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Inverse: If you do not live in Tucson, then you do not live in Arizona.

Truth Value: False

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Converse: If you live in Arizona, then you live in Tucson.

Truth Value: False

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Contrapositive: If you do not live in Arizona, then you do not live in Tucson

Truth Value: True

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Explanation:

The general conditional statement is in the form "If P, then Q". We can write this as "P --> Q" since there is a logical flow or connection from point to point in that exact order.

The inverse of P --> Q is ~P --> ~Q. We negate each piece. To negate something is to do the opposite. If P is true, then ~P is false, and vice versa.

The converse has us swap the positions of P and Q. So the converse would be Q --> P. We don't negate any of the pieces.

Finally, the contrapositive has us do both negating and swapping to get ~Q --> ~P

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Let

P = You live in Tucson

Q = you live in Arizona

Negating each piece has us get

~P = You do not live in Tucson

~Q = You do not live in Arizona

So the inverse would be

~P --> ~Q

If ~P, then ~Q

If you do not live in Tucson, then you do not live in Arizona

This is false. You could live in a city like Phoenix. The fact that you dont live in Tucson doesnt mean you aren't in Arizona.

The original P --> Q statement is true because Tucson is a city in Arizona.

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The converse is Q --> P or "If Q, then P"

Do replacements to get the statement "If you live in Arizona, then you live in Tucson"

This is false. Again you could live in Phoenix or any other city in Arizona and not necessarily in Tucson.

In general, the inverse ~P --> ~Q and the converse Q --> P always have the same truth value.

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Finally the contrapositive.

The format is ~Q --> ~P or "if ~Q, then ~P"

So we get "If you do not live in Arizona, then you do not live in Tucson"

This is a true statement. Imagine throwing a dart at the map. If the dart doesn't hit Arizona at all, then it is impossible to have that same dart hit the city of Tucson.

In general, the original conditional P --> Q and the contrapositive ~Q --> ~P always have the same truth value.