Differentiate between different types of quantifiers with examples

The most usual quantifiers in mathematics are for all (symbolized ∀ - universal quantification) and the existence (symbolize ∃-existential quantifier) and they are not commutative.

Step-by-step explanation:

In logic those two quantifiers the universal and the existential do not commute, in particular in group theory the order defines whether we refer to an inverse or the neutral element of the group, for instance consider the Real numbers with addition, here we say for every real number 'x' there exist a real number 'y' such that x+y=y+x=e. Here e=0. That is the definition of the inverse of x, denote - x.

On the other hand There exist a real number 'e' such that for all real number 'x', we have x+e=e+x=x. Here in the example e=0 and it is called the neutral element of R under addition.

Then in the first example we have ∀x in R ∃ y in R / x+y=y+x = e (Inverse property)

and the second is

∃ e in R ∀ x in R / e+x = x+e = x (Identity property)

Then the first indicate for each one there is some number that fulfills the property of inverse stated, the second says there is one (can be proved it is unique) number that for any real number fulfills the property of identity. Then both quantifiers are clearly different and serve a different purpose.