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26 Cards in this Set
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 Back
Set

A collection of unique elements.


Union

A and B = {x in X: x in A and x in B}


Intersection

A or B = {x in X: x in A or x in B}


Complement

A^(c) = {x in X: x NOT in A)


Set Difference

A\B = A and B^(c) = {x in A and x not in B}


Symmetric Difference

A tri B = (A or B)\(A and B)


Onto (surjective)

Let X,Y be sets and f:X to Y. We say that f maps X onto Y if for each y in Y, there exists a point x in X such that f(x) =y.
in english, all the points in the domain must be hit. 

One to One (injective)

Let X,Y be sets and f:X to Y. We say that f is onetoone if each y in f(X) is mapped to by a unique element of X. Not all y in Y must be hit.


Bijective

both one to one and onto


Domain

the set the function is defined on


Codomain

all the values of the function.


Range

all the values in the Codomain which are mapped to.


Well defined

each point of x maps to exactly one point of Y.


Cardinality

Let A and B be sets. We say that A and B have the same cardinality, card(A) = card(B), if there is a bijection f: A to B.


Cardinality of a finite set

A set A is a finite set with cardinality n if card(A) = card(B) where B = {1,2,3,....,n}.
In other words, the cardinality of set A is the number of elements in 

Cardinality of the empty set

0, no elements in the set


Countable

A set A is countable if card(A) = card(N)
or A is finite 

Uncoutable

otherwise


Bounded above

Let X be an ordered field and let A subset X be nonempty. We say A is bounded above if there is a point M in X such that a<=M for every a in A.


Upper Bound

M is the upper bound


Bounded below

A is bounded below if there is a point m in X such that m<=a for every a in A.


Lower Bound

Here m is the lower bound for A.


Supremum (least upper bound)

Let X be an ordered field and let A subset X be nonempty and bounded above. We say A has a supremum if there is a point in X, denoted sup(A), such that
a. sup(A) is an upper bound for A; b. if b is another upper bound for A, then sup(A)<=b 

Complete

We say that an ordered field X is complete if every set A subset X that is nonempty and bounded above has a least upper bound in X.


Real number system

We define the set R to be the complete ordered field containing Q as an ordered subfield.


Absolute value function

abs(x) = x, x>0
= 0, x = 0 = x, x<0 