1. ## Universal Quantifier Question

Is one of these incorrect?
Also, I often see in proofs that "x in Z" and later they say "but x is arbitrary" is it wrong to use an existential quantifier at the beginning, or is it implied?

1.
(Universal Quantifier) m in Z is even <=> (existential quantifier) k in Z such that 2k = m

2. m in Z is even <=> (existential quantifier) k in Z such that 2k = m

2. $\displaystyle \displaystyle \forall$ means "for all".

So $\displaystyle \displaystyle \forall x \in \mathbf{Z}$ means "for all x that are integers..."

What's wrong with that?

3. So there's no difference between saying:

(universal quantifier) x in Z

x in Z

4. 2. m in Z is even <=> (existential quantifier) k in Z such that 2k = m
This statement has a free variable m, i.e., a variable that is not bound by any quantifier. Generally, formulas with free variables are not propositions, i.e., not something that is either true or false because for each value of a free variable such formula may have its own truth value. For this particular formula, it happens that it is true for all values of m.

Originally Posted by Noxide
Also, I often see in proofs that "x in Z" and later they say "but x is arbitrary" is it wrong to use an existential quantifier at the beginning, or is it implied?
A statement "P(x) holds for an arbitrary x in Z", where P(x) is some expression, is equivalent to "For all x in Z, P(x)".