Chapter 1 Flashcards
Element
A element is an object
Null set
A null set is a set with no elements
Well-defined set
A set is well-define if for each element we can decide if it is in or if it is not in the set
Universal set
All elements under consideration
Complement
Let A be a set; U be the universal set. Ac is everything in the U that’s not in A.
Subsets
Every element of A is also an element of B
Equality
A=B off A is a subset of B and B is a subset of A
Relative complement
A\B
Disjoint
A and B are disjoint if the intersection of A and B is the empty set.
Demorgan’s laws
If A and B are sets, the
a) (AUB)c=Ac and Bc
b) (A and B)c= AcUBc
Functions
Let A and B be sets. A functions, f, from A to B is a rule which associates each element x in A with a unique element f(x) in B.
Domain
Let f:A->B. A is the domain of f
Range
Let f:A->B. The set of elements in B that have some point of A mapped to them by f is called the range of f.
Equal functions
Two functions f and g are equal if
1) domain of f=domain of g
2) f(x)=g(x) for all c in their common domain.
One to one
F is a one to one function if different elements of A are mapping to different elements of B.
If x1,x2 in A with x1 does not equal x2, then f(x1) does not equal f(x2)
If x1, x2 in A with f(x1)=f(x2), then x1=x2
Onto
F is onto if for all y in B, there exist an x in A such that f(x)=y.
“f uses all of B”
Inverse
Let f:X->Y be a one to one function. Then the function f-1:R(f)->X defined by f-1(y)=x provided y=f(x) is call the inverse of the function f.
Field
A field,F, is a non empty set with two operations, + and -, which satisfys the following Binary Associative Commutative Distribution works Identities Additive inverse Multiplication inverse
The order axiom
There is a nonempty subset P(positives) of F for which
1) a,b in P implies a+b is in the positives
2) a,b in P implies a*b is in the positives
3) for any x in F, exactly ONE of the filling holds: x is positive, x is negative , or x is zero.
Order field
A field satisfying the order axiom is an ordered field.
Interval
An interval of real numbers is a set, A, containing at least 2 numbers such that if r,s are in A with r
Upper bound
If there exist a b in R st x
Bounded above
A set is bounded above if it has an upper bound.
Lower bound
If there exist a c in R such that c>=x for all x in A. Then A had a lower bound.
Bounded below
A set is bounded below if it has a lower bound.
Bounded
If a set is bounded above and below.
Unbounded
If it is not bounded
Least upper bound
Let A be a set of real numbers that is bounded above. The number b is called the least upper bound of A if b is an upper bound of A. If k is also an upper bound of A, then b
Supremun
Least upper bound
Greatest lower bound
c is a lower bound of A. If l is a lower bound of A, then c>=l.
Imfimum
Greatest lower bound
Complete
Let S be an ordered field. Then S is complete if for any nonempty subset A of S that is bounded above, lub(A) is in S.
Completeness axiom
R is complete
The Archimedean principle
If a,b are in R with a>0, then there exist an n in the natural numbers such that na >b
Cardinal its
Two sets A and B have the same cardinality if there is a one to one, onto function from the set A to the set B.
Countable
A set is countable if it has the same cardinality as some subset of the natural numbers
Set
A set is a collection of objects
Cartesian product
Let A1,A2,…An be a finite collection of sets. The Cartesian product of the collection is A1xA2x…xAn={(a1,a2,….,an)|aj is in Aj for j=1,2,3….,n}
Ex: A={1,2,4} and B={3,5,6}
The AxB={(1,3),(1,5),(1,6),(2,3),(2,5), ect
Triangle inequality
|a+b|
