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13397139
Basics of Set Theory: Natural and Real Numbers
Description
Senior Freshman Mathematics Mind Map on Basics of Set Theory: Natural and Real Numbers, created by Luke Byrne on 22/04/2018.
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natural numbers
natural
numbers
real numbers
real
set theory
additive
multiplicative
mathematics
senior freshman
Mind Map by
Luke Byrne
, updated more than 1 year ago
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Luke Byrne
over 6 years ago
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Resource summary
Basics of Set Theory: Natural and Real Numbers
"Set theory started around 1870’s → late development in mathematics but now taught early in one’s maths education due to the Bourbaki school."
A set is a collection of objects. x ∈ A means the element x is in the set A (i.e. belongs to A).
e.g. all students in a class
N the set of natural numbers starting at 0
0 ∈ N
if x ∈ N, then x + 1 ∈ N (x ∈ N → x + 1 ∈ N)
R is the set of real numbers. It is governed by the following axioms:
CLOSURE
Additive
∀x, y ∃z(x + y = z)
Multiplicative
∀x, y, ∃z(x × y = z)
Associativity
Additive
∀x, y, z: x + (y + z) = (x + y) + z
Multiplicative
∀x, y, z: x × (y × z) = (x × y) × z
Commutativity
Additive
∀x, y: x + y = y + x
Multiplicative
∀x, y: x × y = y × x
Distributivity
∀x, y, z: x × (y + z) = (x × y) + (x × z) and (y + z) × x = (y × x) + (z × x)
Identity
Additive
There is a number, denoted 0, such that for all x, x + 0 = x.
Multiplicative
There is a number, denoted 1, such that for all x, x * 1 = 1 * x = x.
Inverses
Additive
For every x there is a number, denoted -x, such that x + (-x) = 0
Multiplicative
For every nonzero x there is a number, denoted x^−1, such that (x * x^-1) = (x^-1 * x) = 1.
0 != 1
Irreflexivity of <
~(x < x)
Transitivity of <
If x < y and y < z, then x < z
Trichotomy
Either x < y, y < x, or x = y
Completeness
If a nonempty set of real numbers has an upper bound, then it has a least upper bound.
If x < y, then x + y < y + z.
If x < y and 0 < z, then x * z < y * z and z * x < z * y.
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