Thread: Analysis question on quantifiers

Something really important that I have learned is that the order in which one writes the quantifiers is of paramount importance; writing them in the wrong order can affect the truth value.

Now here's my question: Am I looking at quantifiers in a correct way? Let me give an example:



This says that for all possible x values, there exists a y. For every x value, there is at least one y that satisfies whatever..

Then you have:



This says: some y value such that for every possible x... whatever. To me, this makes y appear more "fixed" than in the first statement, since in the first statement, the y is in the context of all x.

Finally, I have encountered statements like:



How must I look at such a statement?

I have posted a lot of questions about this topic, and I feel like I'm beating a dead horse. However, I really want to ensure that I understand this topic well. I've never taken a class like this before, it seems like a whole new way of thinking.

Thanks

Hello Skinner

Originally Posted by Skinner

This says that for all possible x values, there exists a y. For every x value, there is at least one y that satisfies whatever..

Correct. For example, if x is a real number, then for all x, there exists a y such that y > x. In other words, there is no largest real number.

Originally Posted by Skinner

This says: some y value such that for every possible x... whatever. To me, this makes y appear more "fixed" than in the first statement, since in the first statement, the y is in the context of all x.

Again, I think you are interpreting this correctly; the value of y is chosen before the values of x. For example, if x and y are integers, there exists a y such that for all x, xy = x. (The integer is, of course, y = 1.)

Originally Posted by Skinner

How must I look at such a statement?

This is rather harder to come up with a simple example. But you just combine the ideas in the first two examples; it means, of course, what it says, reading from left to right: For all values of x, there exists a value of y such that, for all values of z, such-and-such a statement is true. The following example works, but it may seem a little trivial.

Suppose we are considering elements of the set of non-negative integers: . Then



Can you see what this value of is?

Does that help?

Grandad

Take care that in this example, when you chose , this choice seems independent from the choice of (and indeed is also a solution for )

An exemple where the permutation is impossible:



given an works, but there is no in such that for any

I was checking this example:




If there exists a fixed y value for every z value, isn't false? If so, isn't that implication statement always true?

Isn't y also an element of if y is an element of ?

My mathematical knowledge is very limited, but I thought this would have been true.

If there exists a fixed y value for every z value, isn't false? If so, isn't that implication statement always true?

Well, if you fix an integer , there is a infinite amount of real numbers such that

Maybe what you've read is which is always true because of what you said. But in my formula, the whole implication is under the -scope.

Originally Posted by clic-clac

Well, if you fix an integer , there is a infinite amount of real numbers such that

Sure, if you put those restrictions on . But if , that's false, right?

Regarding your formula,

You said this was different from what I read: and I understand why you said that. So how must I look at your formula?

This relates to the fact that I'm trying to understand how the order of the the quantifiers changes the formula.

Yeah, " -formulas " begin to be complex! is false for real numbers, I agree: there is no real number greater than all the others.

Let's see :

Take a real number, for example .

The formula says that there exists a real number which is an integer and such that: So is there such a ? What about ?

Let be a real number. If then . The formula works for . But if you take any other real number for , I guess you see that you can find an integer such that what we want. In fact, any integer lower than is a solution; but of course, it will depend on ( is a solution for but not for ).

Hope you see better the meaning of the formula.

OH! Okay, I see it now!

This really helps me to understand what the problem itself is asking me. Okay, for any , we can pick out an integer . There is always an integer for every . If is less than that integer , is not necessarily less than , making the implication false.