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26 Cards in this Set

  • Front
  • Back
A collection of unique elements.
A and B = {x in X: x in A and x in B}
A or B = {x in X: x in A or x in B}
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 one-to-one if each y in f(X) is mapped to by a unique element of X. Not all y in Y must be hit.
both one to one and onto
the set the function is defined on
all the values of the function.
all the values in the Codomain which are mapped to.
Well defined
each point of x maps to exactly one point of Y.
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
A set A is countable if card(A) = card(N)
A is finite
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
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