Quantifiers

All mathematical quantifiers can be expressed in either there exists or for all.

There exists

• 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

• 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)]