LinkBack URL
About LinkBacksDetermine 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
Originally Posted by questionboy
Seems to be you want to check whether.
Well, letbe 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,
If I didn't commit some logical mistake above () then the answer is: no, they aren't logical equivalent.
Tonio
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.
Thank you very much!
I am wondering why this is false
means: for any integer x we have
for any integer x,
, which is false
Because it is not true that for any integer |x|=x iff for every integer x, x>=0...for example , |7|=7 but it is NOT true that x >=0 for every integer...say, -1 < 0...
Tonio
  |
