13397139
Description
Mind Map by Luke Byrne, updated more than 1 year ago
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Created by Luke Byrne
over 6 years ago
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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)
- 0 ∈ N
- e.g. all students in a class
- A set is a collection of objects. x ∈ A means the element x is in
the set A (i.e. belongs to A).
- R is the set of real numbers. It is governed by the following axioms:
- CLOSURE
- Additive
- ∀x, y ∃z(x + y = z)
- ∀x, y ∃z(x + y = z)
- Multiplicative
- ∀x, y, ∃z(x × y = z)
- ∀x, y, ∃z(x × y = z)
- Additive
- Associativity
- Additive
- ∀x, y, z: x + (y + z) = (x + y) + z
- ∀x, y, z: x + (y + z) = (x + y) + z
- Multiplicative
- ∀x, y, z: x × (y × z) = (x × y) × z
- ∀x, y, z: x × (y × z) = (x × y) × z
- Additive
- Commutativity
- Additive
- ∀x, y: x + y = y + x
- ∀x, y: x + y = y + x
- Multiplicative
- ∀x, y: x × y = y × x
- ∀x, y: x × y = y × x
- Additive
- Distributivity
- ∀x, y, z: x × (y + z) = (x × y) + (x × z) and
(y + z) × x = (y × x) + (z × x)
- ∀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.
- 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.
- There is a number, denoted 1, such that for all x, x * 1 = 1 * x = x.
- Additive
- Inverses
- Additive
- For every x there is a number, denoted -x, such that x + (-x) = 0
- 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.
- For every nonzero x there is a number,
denoted x^−1, such that (x * x^-1) = (x^-1 * x) =
1.
- Additive
- 0 != 1
- Irreflexivity of <
- ~(x < x)
- ~(x < x)
- Transitivity of <
- If x < y and y < z, then x < z
- If x < y and y < z, then x < z
- Trichotomy
- Either x < y, y < x, or x = y
- 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 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.
- If x < y and 0 < z,
then x * z < y * z
and z * x < z * y.
- CLOSURE
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