- #1
lpau001
- 25
- 0
Hey! I tried to make the title as descriptive as possible, but ran out of characters. I am trying to prove that..
"There exists x [itex]\in[/itex] (1, [itex]\infty[/itex]) such that for all y [itex]\in[/itex] (0,1), xy[itex]\geq[/itex]1.
[itex]\exists[/itex] x [itex]\in[/itex] (1, [itex]\infty[/itex]) s.t. [itex]\forall[/itex] y [itex]\in[/itex] (0,1), xy[itex]\geq[/itex]1.
none.
I say 'false' because when the entire statement is negated, the working negation is true.
[itex]\neg[/itex]{[itex]\exists[/itex] x [itex]\in[/itex] (1, [itex]\infty[/itex]) s.t. [itex]\forall[/itex] y [itex]\in[/itex] (0,1), xy[itex]\geq[/itex]1.} (Negating line)
[itex]\forall[/itex] x [itex]\in[/itex] (1, [itex]\infty[/itex]), [itex]\exists[/itex] y [itex]\in[/itex] (0,1), s.t. xy < 1. (This is the working negation of original statement)
Now looking at this statement, since x can be infinitely large, and I can pick an infinitely smaller y, the negation would be true, making the original statement false.
But if I look at the original statement, can't I do the same thing? Would this be a paradox?
Also, there is a similar problem, except the original statement is "For all y's in the element (0,1) there exists an x in the element (1, infinity) such that xy < 1 ." I get the same result, except in this one, since x and y can get infinitely close to 1, albeit on either side, they will cancel each other out, making the working negation true, and the statement false.
Homework Statement
"There exists x [itex]\in[/itex] (1, [itex]\infty[/itex]) such that for all y [itex]\in[/itex] (0,1), xy[itex]\geq[/itex]1.
[itex]\exists[/itex] x [itex]\in[/itex] (1, [itex]\infty[/itex]) s.t. [itex]\forall[/itex] y [itex]\in[/itex] (0,1), xy[itex]\geq[/itex]1.
Homework Equations
none.
The Attempt at a Solution
I say 'false' because when the entire statement is negated, the working negation is true.
[itex]\neg[/itex]{[itex]\exists[/itex] x [itex]\in[/itex] (1, [itex]\infty[/itex]) s.t. [itex]\forall[/itex] y [itex]\in[/itex] (0,1), xy[itex]\geq[/itex]1.} (Negating line)
[itex]\forall[/itex] x [itex]\in[/itex] (1, [itex]\infty[/itex]), [itex]\exists[/itex] y [itex]\in[/itex] (0,1), s.t. xy < 1. (This is the working negation of original statement)
Now looking at this statement, since x can be infinitely large, and I can pick an infinitely smaller y, the negation would be true, making the original statement false.
But if I look at the original statement, can't I do the same thing? Would this be a paradox?
Also, there is a similar problem, except the original statement is "For all y's in the element (0,1) there exists an x in the element (1, infinity) such that xy < 1 ." I get the same result, except in this one, since x and y can get infinitely close to 1, albeit on either side, they will cancel each other out, making the working negation true, and the statement false.