All mathematical quantifiers can be expressed in either there exists or for all.
- There exists an object x having property P.
- Denoted by symbol - ∃
- For example ∃x[x2+2x+1=0]
- You can prove it by finding x which satisfies the property.
- ∃! - There exists an unique.
Continuous function - Graph of the function does not have any breaks or joints
- For all x it is the case that.
- Denoted by symbol - ∀
- For example - ∀x(x2 >= 0).
How to read mathematical formulas
- (∀x)[P(x) => Q(x)] : For every x, if P(x) then Q(x).
- (∀x)[P(x) ^ Q(x)] : ∀xP(x) ^ ∀xQ(x) : For every x P(x) and Q(x).
- (∃x)[P(x) ^ Q(x)] : There is an x for which P(x) and Q(x).
- (∃x)[P(x) => Q(x)] : There is an for which if P(x) then Q(x). Rarely useful.
Negation of quantifiers.
¬(∀x[P(x) => Q(x)]) = ∃x[P(x) !=> Q(x)] = ∃x[P(x) ^ ¬Q(x)]