Determine whether All X ( P(X) If only if Q(X)) and All X P(X) If only if All X Q(X) are logically equivalent.

Thank you very much

Printable View

- March 23rd 2010, 06:48 PMquestionboylogically equivalent
Determine whether All X ( P(X) If only if Q(X)) and All X P(X) If only if All X Q(X) are logically equivalent.

Thank you very much - March 23rd 2010, 07:42 PMtonio

Seems to be you want to check whether .

Well, let be the universe from where we get our objects ,and let P(x) = the absolute value of x is x itself (i.e., P(x) iff ,and let Q(x)= x is a non-negative integer (i.e., Q(x) iff ) , then:

means: for any integer x, , which is true, whereas

means: for any integer x we have for any integer x, , which is false.

If I didn't commit some logical mistake above ((Giggle)) then the answer is: no, they aren't logical equivalent.

Tonio - March 23rd 2010, 07:44 PMwgunther
No. Consider P(x) = "x=0", and Q(x)="x>0"

Clearly For all x, P(x) iff Q(x) is false. But, "for all x, P(x)" is false, and "for all x, Q(x)" is false. Then "for all x, P(x) iff for all x, Q(x)" is true.

The converse is true though, because assume "for all x, P(x) iff Q(x)". Then if there there is a x where P(x) is false then Q(x) is also false, so "for all x, P(x) iff for all x, Q(x)" is true as each universal is false. Otherwise, P(x) is always true, as is Q(x); result follows. - March 23rd 2010, 07:59 PMquestionboy
Thank you very much!

I am wondering why this is false

http://www.mathhelpforum.com/math-he...76a15151-1.gif means: for any integer x we have http://www.mathhelpforum.com/math-he...b9710395-1.gif for any integer x, http://www.mathhelpforum.com/math-he...baa89612-1.gif , which is false - March 24th 2010, 01:30 AMtonio