Contrapostive Confusion: Resolving Necessary & Sufficient Conditions

[mr_coffee]

More logic fun! contrapostive confusion, book says one thing, and yet...:Resolved:.

Hello everyone!

I'm quite confused on this issue. The directions say:
Use the contrapositive to rewrite the statements in 45 and 46 in "if-then" form in two ways.

So i first look back to see what contrapositive means. The book says:
The contrapositive of a conditional statement of the form "if p then q" is
if ~q then ~p.
Symbolically, the contrapostive of p-->q is ~q--> ~p.

I then look at the problem. Problem 45 has the answer in the back of the book.

#45. Being divisble by 3 is a necessary condtion for this number to be dividble by 9.

answer:
If this number is not divisble by 3, then it is not divisible by 9.

If this number is divisble by 9, then it is divisble by 3.

When i look at this, it doesn't follow the contrapostive form, but instead follows Necessary and Sufficient Conditions.

if r and s are statements:
r is a necessary condtion for s, mans "if not r then not s."
r is a necessary conditon for s also means "if s then r."

Why did they say contrapostive if they are not using it?


I did #46 this way:
Doing homework regularly is a necessary condtion for Jim to pass the course.

#my answer:
If Jim does not do homework regulary, then he will not pass the course.

If Jim is to pass the course, then he must do homework regularly.

Do you think that is correct? or am I not allowed to use Jim twice?

Should it be:

If not doing homework regularythen he will not pass the course.

If Jim is to pass the course, then he must do homework regularly.

Thanks!

 

[0rthodontist]

answer:
If this number is not divisble by 3, then it is not divisible by 9.

If this number is divisble by 9, then it is divisble by 3.

When i look at this, it doesn't follow the contrapostive form, but instead follows Necessary and Sufficient Conditions.

if r and s are statements:
r is a necessary condtion for s, mans "if not r then not s."
r is a necessary conditon for s also means "if s then r."

They are using the contrapositive implicitly when they give this definition. "if not r then not s" is equivalent to--is the contrapositive of--"if s then r." The definition of necessary and sufficient conditions in your book follows the contrapositive.

If Jim does not do homework regulary, then he will not pass the course.

If Jim is to pass the course, then he must do homework regularly.

I would call this correct. It is a little informal with the way you use the terms "must," "will," and "does," but it gets the point across and blander language doesn't really sound right. The use of "Jim" isn't a problem. It's more grammatical than the alternative you mentioned, though if you said "If Jim is not doing homework regulary then he will not pass the course" that would also be correct.

 

[mr_coffee]

Excellent! thanks again!