# Week 3 Mathematical Thinking

### 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[x
^{2}+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(x
^{2}>= 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)]