1 - Numbers Flashcards
Foundation of natural numbers, integers, rational, real, and complex numbers. Incompleteness of rationals and completeness of reals. Infimum, supremum.
What is a supremum?
A real number α is called the least upper bound (or supremum, or sup) of S, if:
(i) α is an upper bound for S; and
(ii) there does not exist an upper bound for S that is strictly smaller than α. The supremum, if it exists, is unique, and is denoted by sup S.
α = sup S ⇐⇒ (i) x ≤ α for all x ∈ S, and (ii) for every ε > 0 there exists x ∈ S such that x > α − ε.
What is an infimum?
A real number α is called the greatest lower bound (or infimum, or inf) of S, if:
(i) α is a lower bound for S; and
(ii) there does not exist a lower bound for S that is strictly larger than α. The infimum, if it exists, is unique, and is denoted by inf S.
α = inf S ⇐⇒ (i) x ≥ α for all x ∈ S, and (ii) for every ε > 0 there exists x ∈ S such that x < α + ε.
What is the Completeness Axiom?
Every non-empty subset of Real numbers which is bounded from above has a least upper bound, or namely a supremum.
In other words, the Completeness Axiom guarantees that, for any nonempty set of real numbers S that is bounded above, a supremum exists (in contrast to the max, which may or may not exist. An analogous property holds for inf S: Any nonempty subset of R that is bounded below has a greatest lower bound.
What are the properties and operations of the set of Natural Numbers?
- Addition: a + b
- Associativity: a + ( b + c ) = ( a + b ) + c
- Commutativity: a + b = b + a
- Monotone Property: a > b => a + c > b + c
- Multiplication: a * b
- Associativity: a * ( b * c ) = ( a * b ) * c
- Commutativity: a * b = b * a
- Monotone: a > b –> a * c > b * c
- Distributive Property: a * ( b + c ) = ( a * b ) + ( a * c )
What is the Archimedean Property?
Given any real number x, there exists n ∈ |N such that n > x.
In other words, this says that the set of natural numbers is not bounded (from above).
Or as old mate Qui-Gon puts it, “There’s always a bigger fish”
What is an upper bound?
A real number α is called an upper bound for S if x ≤ α for all x ∈ S. The set S is said to be bounded above if it has an upper bound.
What is a lower bound?
A real number α is called a lower bound for S if x ≥ α for all x ∈ S. The set S is said to be bounded below if it has a lower bound.
What is the set N?
N is the set of natural numbers. All positive integers:
{1,2,3,4,…}
What is the set Z?
Z is the set of integers, ie. positive, negative or zero:
{…,-3,-2,-1,0,1,2,3,…}
The set N is included in the set Z (because all natural numbers are part of the relative integers).
Z∗ (Z asterisk) is the set of integers except 0 (zero).
What is the set Q?
Q is the set of rational numbers, Where all solutions of Q take the form:
Q ϵ { a/b | a ϵ Z, b ϵ N }
What is the set R?
R is the set of real numbers, ie. all numbers that can actually exist, including rational numbers, non-rational numbers or irrational numbers such as π or √2.
What is the set C?
C is the set of complex numbers, ie. the set of real numbers R and all imaginary numbers I.
Taking the form:
C ϵ { x+yi | x,y ϵ R }
What does |A| mean?
|A|, called cardinality of A, denotes the number of elements of A. For example, if A={(1,2),(3,4)}, then |A|=2.
What does A=B mean?
A=B if and only if they have precisely the same elements. For example, if A={4,9} and B={4,9}, then A=B.
What does A⊆B mean?
A⊆B if and only if every element of A is also an element of B. We call A a subset of B. For example, {1,8,1107}⊆N.
This means that A is either a subset of B or it is equal to B.
What does a∈A mean?
a∈A means a is a member of A. For example, 5∈Q
What does a∉A mean?
a∉A means a is not a member of A. For example, 2.7∉Z
What does A∩B mean?
A∩B denotes the set containing elements that are in both A and B. A∩B is called the intersection of A and B. For example, if A={1,2} and B={2,3}, then A∩B={2}.
What does A∪B mean?
A∪B denotes the set containing elements that are in either A or B or both. A∪B is called the union of A and B. For example, if A={1,2} and B={2,3}, then A∪B={1,2,3}.
What does A∖B mean?
A∖B denotes the set containing elements that are in A but not in B. A∖B is read as “A drop B”. For example, if A={1,2} and B={2,3}, then A∖B={1}.
What is ∅?
∅ denotes the empty set, the set with no members.
What are axioms?
A statement which is regarded as being established, accepted, or self-evidently true. A postulate.
