does anyone know how to prove this statement? ( algebra )?

Question

Universal quantifiers x , universal quantifiers y, ( x>=y)

what does it mean?
does it mean that every real number is as large as any real number?
this statement is false , isn't is?

do you know how to prove it?
help me please..

thanks

AxAy, (x>=y)


actually the "A" is upside down

Answer 1

∀ x ∀ y (x ≥ y) is false, obviously. That would be saying that every number is bigger than every other number. As a counterexample, let x = 1 and y = 2.

∀ x ∃ y (x ≥ y) is true. Every number has a number larger than itself.

As a proof, suppose that it's not true. Then let x = y + 1, a contradiction.

Answer 2

As you state it, it sounds more like part of the statement of a theorem, because it is not true in general. Actually, it says that for all x and y, x is at least as large as y. A trivial example will be sufficient to disprove it, such as x = 1, y = 2.

Are you certain that it wasn't supposed to be an existential quantifier in front of y? Then again that statement would be self-evident.

Answer 3

The upside down 'A' means for each, it means For each Y there exists X >= Y. It means for every number, there is a larger number. I forget the proof for that off the top of my head, but it's out there, check Wikipedia.

Answer 4

As stated, this is false. As stated, it reads "for every x, for every y, x>=y". But take x=1 and y=2 so that x
If the statement was "for every x there exists a y such that x>=y", then it would be true. (The symbol for "there exists" is a backwards capital "E".)

If the statement was "there exists x and there exists y such that x>=y", then the statement is again true.

If the statement was "there exists x such that for every y, x>=y", then the statement is false.

Are you sure you have the problem statement correct?

Answer 5

To prove it false, you need only display a counterexample. The statement is claiming that for all x and for all y, x >=y.

A counterexample is very easy in this case: For example, just let the domain of definition be integers, let x be 2 and let y be 3. This is an example where the claim is false. There are many other counterexamples as well.

Answer 6

It's false, as it would mean that every real number is less than every other real number.

However, AxEy(x>=y) is true (where A is upside down and E is backwards).

Answer 7

the value of x must be greater then or equal to the value of y.

so if the value of y is 2
x must be great then 2 or equal to 2.

It falls under the element of the reals catagory to account for all numbers, including whole and fractions. (2.00000001 is considered greater then 2)
I'm not sure though if real numbers include irrational or not.

Answer 8

Upside down A means "For all."

Answer 9

that is not true. it could be "for any x, there exists y such that x>=y"