Intereting Posts

Summing $ \sum _{k=1}^{n} k\cos(k\theta) $ and $ \sum _{k=1}^{n} k\sin(k\theta) $
Derivation of the polarization identities?
Field extensions and algebraic/transcendental elements
Show that minimum exists (direct method)
Why do we use groups and not GROUPS?
why 64 is equal to 65 here?
Not getting $-\frac{\pi}{4}$ for my integral. Help with algebra
Smooth classification of vector bundles
Toss a fair die until the cumulative sum is a perfect square-Expected Value
Why do the elements of finite order in a nilpotent group form a subgroup?
Probability that two randomly chosen permutations will generate $S_n$.
Given a $4\times 4$ symmetric matrix, is there an efficient way to find its eigenvalues and diagonalize it?
Notation for: all subsets of size 2
question about the limit $\lim_{h\to0}\frac{\arcsin(x+h)-\arcsin(x)}{h}$
Projective closure of an algebraic curve as a compactification of Riemann surface

What’s the difference between $\forall \space x \space \exists \space y$ and $\exists \space y \space \forall \space x$ ? I don’t believe they mean the same thing even though the quantifiers are attached to the same variable, but I’m having a hard time understanding the difference. Any examples to make the distinction clear would be appreciated.

- Prove or disprove validity: $(\forall x \exists y (P(x) \supset Q(y))) \supset(\exists y \forall x (P (x) \supset Q(y)))$
- Consistency of Peano axioms (Hilbert's second problem)?
- Who first discovered that some R.E. sets are not recursive?
- Logical issues with the weak law of large numbers and its interpretation
- Is the null set a subset of every set?
- Could someone show me a simple example of something being proved unprovable?
- Counting Rows of a Truth Table that Satisfy a Condition
- Can we prove that odd and even numbers alternate without using induction?
- Axiom of choice, non-measurable sets, countable unions
- Difference between “for any” and “for all”?

Consider this example.

For all $x\neq 0$ there is a $y\neq 0$ such that $xy = 1$.

There is a $y\neq 0$ such that for all $x\neq 0$ you have $xy = 1$.

You can probably see that the one statement is true and the other false.

Every natural number has a successor. There is no natural number which is the successor of *every* number.

One reads “For every $x$, there exists a $y$…”, and the other says “There exists a $y$, such that for every $x$…”

An example of the difference can be found by making the (totally non-mathy) statement:

$$\forall x \;\exists y \text{ s.t. $x$ loves $y$}$$

That is, everybody loves at least one other person.

On the other hand:

$$\exists y \;\forall x \text{ s.t. $x$ loves $y$}$$

That is, there is a person that everyone loves.

Does this make the difference a bit more clear?

$\forall m \exists k | k>m$ which in plain English means: For any integer, there

is another integer greater than it.

$\exists k \forall m | k>m$ which in plain English means: There is some integer

that is greater than every integer.

The only difference is the order of the quantifiers, but the meaning

is MUCH changed. In fact, the first is true and the second is false. I

am NOT saying that one is the correct order and one is the incorrect

order. They are just statements that say different things.

source : http://mathforum.org/library/drmath/view/61813.html

Everybody has a nose, but there is no nose that belongs to everybody.

A quick summary of the essential parts of what others have written: $\forall x\,\exists y$ allows $y$ to depend on $x$; $\exists y\,\forall x$ requires the same $y$ for all $x$’s.

You really should read some of Daniel Velleman’s *How to Prove It,* 2nd ed. He discusses this kind of stuff in the beginning of his chapter 2. (Chapter 1 is short, covering sentential logic.)

There has been a lot of clear examples given as answers.

So, not giving any example but trying to make a stronger sense:

You can regard the quantifiers part of your formula as a *bound*. Therefore, comparing $\forall x \exists y$ vs $\exists x \forall y$, you should consider the first one as “first bound your formula by $x$ and **then** relative to $x$, bound it by $y$”. For the second one, the matter must be obvious(interchange $x$ and $y$ in the quotation marks). Consequently, these two different ways of bounding are not necessarily the same.

One is, “for each x there is a y, the other, “there is a (single) y for all x.” It is a “uniformity difference”.

As others have pointed out one says “for all x, there exists a y such that…” while the other says “there exists a y, such that for all x…”

An operation O on a finite number of elements can get described by its operation table:

```
O a b c ...
k
j
l
.
.
.
```

with the points where horizontal and vertical lines intersect describing the value of O(x, y) at point (x, y).

Thus, if we have a “for all x, there exists a y such that O(x, y)=z” statement, that means for any input from k, j, l… there exists a single input from a, b, c … such that O(x, y)=z. A “there exists a y such that for all x, O(x, y)=z” statement means that for a single input from a, b, c, … all elements from the k, j, l … satisfy O(x, y)=z.

To understand this a little more concretely consider this operation table

```
A 0 1 2
0 2 0 1
1 1 0 2
2 1 0 2
```

We have that for all x, there exists a y such that A(x, y)=2. Here, if x=0, there exists the y=0, such that A(x, y)=2, if x=1, there exists y=2 such that A(x, y)=2, and if x=2, then y=2 such that A(x, y)=2. Similarly, if you have a “for all x, there exists a y such that O(x, y)=z” statement concerning a binary operation, then for each row of its operation table there will exist an element y such that x and y will intersect at element z. Thus, z will appear at least once in each row. Symmetrically, if we have a “for all y, there exists an x such that O(x, y)=z” statement, then z will appear at least once in each column.

On the other hand, for operation A we have that there exists a y, such that for all x, A(x, y)=0. In this case, y=1. Similarly, if you have a “there exists a y, such that for all x, O(x, y)=z” statement concerning a binary operation, then its operation table will have a single element z which appears throughout at least one column. In a symmetrical manner, if we have a “there exists an x, such that for all y”, then its operation table will have single element z which throughout each row. See the difference?

- Boy and Girl paradox
- Inequality between 2 norm and 1 norm of a matrix
- Question about complete orthonormal basis
- A Proof of Legendre's Conjecture
- Bijective local isometry to global isometry
- What's the difference between a contrapositive statement and a contradiction?
- How to show this presentation of the additive group $(\mathbb{Q},+)$?
- Properties of compact set: non-empty intersection of any system of closed subsets with finite intersection property
- How to determine pointwise limit/uniform convergence.
- Medial Limit of Mokobodzki (case of Banach Limit)
- “Conic sections” that are really just two straight lines
- Difference between orthogonal projection and least squares solution
- Proof of Sobolev Inequality Theroem
- Limit of nth root of n!
- A function can provide the complete set of Euler primes via a Mill's-like constant. Is it useful or just a curiosity?