# Math Help - Express in words from symbols

1. ## Express in words from symbols

Can anyone help with the following:

Express in words the meaning of ∃xP(x) ∧ ∀x∀y ((P(x) ∧ P(y)) → (x = y))

Here is what I came up with:

For some P(x), and for every x and y, if P(x) and P(y), then x=y.

But this doesn't make too much sense to me.

2. ∃xP(x) just means there exists some x for which the property P holds.
The following just means that there is only one element for which P holds.

3. Originally Posted by kturf
Can anyone help with the following:

Express in words the meaning of ∃xP(x) ∧ ∀x∀y ((P(x) ∧ P(y)) → (x = y))

Here is what I came up with:

For some P(x), and for every x and y, if P(x) and P(y), then x=y.

But this doesn't make too much sense to me.
I find this statement strange. I haven't taken any formal logic classes but the operators are all common ones in math.

I too think it means that for all x and y where P(x) and P(y) are true (maybe exist?) then x=y, implying that P(x)=P(y) implies x=y. I don't know if this is correct because many of the operators have multiple uses in different contexts. The main part that confuses me is the AND operator, ^. It could mean something else, lattice theory for example, which is why I find it strange. Also, y was never defined as an element but x was.

I hope someone will explain this in detail for all of us...

4. Originally Posted by kturf
Can anyone help with the following:

Express in words the meaning of ∃xP(x) ∧ ∀x∀y ((P(x) ∧ P(y)) → (x = y))

Here is what I came up with:

For some P(x), and for every x and y, if P(x) and P(y), then x=y.

But this doesn't make too much sense to me.
You can make any statement for the proposition P(x).

For ∃xP(x), I can say there is at least a member, x, whose image is P(x).

For ∀x∀y ((P(x) ∧ P(y)) → (x = y)), I can say for every two images who possess the same appearance, they are the same images of the same person, i.e, x and y are of the same person.

This reminds me of the one-one function.

Say: There is at least an x in the domain (meaning one or more). If for every x and y, f(x)=f(y), then x = y.

P(x)^P(y) $\rightarrow$ x=y could mean if x is P(x) and y is P(y), then x is y.