∃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.
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 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...
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) x=y could mean if x is P(x) and y is P(y), then x is y.