Real quiz 2

one to one / injective

A function f: X→ Y is if for every x1, x2 in X when f(x1) = f(x2) then x1=x2

onto / surjective

A function f:X→ Y is if for any y in Y there exists x in X s.t. f(x)= y

Same cardinality

A set A has the same cardinality as B if there exists f:A→ B that is one to one and onto (A~B)

countable

A set A is countable if it is either finite or if A~N. In the case than A~N, we say A is countably infinite

Nested Interval Property

If {In} inf n=1 is a nested sequence of nonempty bounded closed intervals, then {In} inf n=1 does not equal the empty set

Archimedean Property

1. Given any number x in R, there exists an n in N satisfying n >x

2. Given any real number y>0, there exists an n in N satisfying 1/n<y

Theorem 1.5.7

If B is a countable set and A is a subset of B, then A must either be finite or countable

Theorem 1.5.8

The union of a finite or countable collection of countable sets is always a countable set

N

countable

Z

countable (alternative bijection)

Q

countable (sets of p and q adding to a number)

R

uncountable