Here is a short list of different mathematical notations commonly used. Mathematicians use these to make proofs more compact and clearer (plus, it guarantees them that they still have jobs given the fact that the field Mathematics is centuries old :P ). While it may be difficult to get used to at first, these can make writing answers quicker, which may help on exams...
Contents
Special Sets
The following list is the shorthand way to describe several special sets (sets are simply a collection of numbers).
$ \,\mathbb{N}\, $ denotes the set of all natural numbers, i.e. {1, 2, 3, ...}
$ \,\mathbb{Z}\, $ denotes the set of all integers, i.e. {..., -2, -1, 0, 1, 2, ...}
$ \,\mathbb{Q}\, $ denotes the set of all rational numbers, i.e. all numbers that can be written as a ratio of two integers
$ \,\mathbb{R}\, $ denotes the set of all real numbers, i.e. any number without $ \,j\, $
$ \,\mathbb{C}\, $ denotes the set of all complex numbers, i.e. numbers of the form $ \,a+bj\, $, this includes rational numbers
Is a Element of
The symbol $ \,\in \, $ is read as "is a element of". It is used to describe a variable as a an element of a set. Examples include:
$ \,x\in \mathbb{R}\, $ is the shorthand way of saying "x is a rational number"
$ \,s,t\in \mathbb{Z}\, $ is the shorthand way of saying "s and t are integers"
There Exists
The symbol $ \,\exists \, $ is read as "there exists". It is used to say that there exists a value that satisfies some condition. Examples include:
$ \,\exists t\in \mathbb{Z}\, $ is the short hand way of saying "there exists an integer t"
For All
The symbol $ \,\forall \, $ is read as "for all". It is used to say that a condition/ result applies for all elements in a set. Examples include:
$ \,\forall c\in \mathbb{Q}\, $ is the short hand way of saying "for all rational numbers c"
Some Commonly Used Notations
$ \,\exists \epsilon \in \mathbb{R}\, $ such that $ \,|x(t)|<\epsilon , \forall t\in \mathbb{R}\, $ is read as "there exists some real number epsilon such that the function $ x(t) $ is bounded above by positive epsilon and below by negative epsilon, for all real values of t" or simply x(t) is bounded for all t.
There is no $ \,\delta \in \mathbb{R}\, $ such that $ \,|y(t)|<\delta , \forall t\in\mathbb{R}\, $ is read as "there is no real number delta such that the function y(t) is bounded above by positive delta and below by negative delta for all real values of t" or simply y(t) is unbounded.
Examples Including the Above Notations
(Anyone, feel free to add any others)
HW2.A Jeff Kubascik_ECE301Fall2008mboutin
HW2.C Jeff Kubascik_ECE301Fall2008mboutin
