Basics of Set Theory: Natural and Real Numbers

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Senior Freshman Mathematics Mind Map on Basics of Set Theory: Natural and Real Numbers, created by Luke Byrne on 22/04/2018.
Luke Byrne
Mind Map by Luke Byrne, updated more than 1 year ago
Luke Byrne
Created by Luke Byrne over 6 years ago
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Resource summary

Basics of Set Theory: Natural and Real Numbers
  1. "Set theory started around 1870’s → late development in mathematics but now taught early in one’s maths education due to the Bourbaki school."
    1. A set is a collection of objects. x ∈ A means the element x is in the set A (i.e. belongs to A).
      1. e.g. all students in a class
        1. N the set of natural numbers starting at 0
          1. 0 ∈ N
            1. if x ∈ N, then x + 1 ∈ N (x ∈ N → x + 1 ∈ N)
        2. R is the set of real numbers. It is governed by the following axioms:
          1. CLOSURE
            1. Additive
              1. ∀x, y ∃z(x + y = z)
              2. Multiplicative
                1. ∀x, y, ∃z(x × y = z)
              3. Associativity
                1. Additive
                  1. ∀x, y, z: x + (y + z) = (x + y) + z
                  2. Multiplicative
                    1. ∀x, y, z: x × (y × z) = (x × y) × z
                  3. Commutativity
                    1. Additive
                      1. ∀x, y: x + y = y + x
                      2. Multiplicative
                        1. ∀x, y: x × y = y × x
                      3. Distributivity
                        1. ∀x, y, z: x × (y + z) = (x × y) + (x × z) and (y + z) × x = (y × x) + (z × x)
                        2. Identity
                          1. Additive
                            1. There is a number, denoted 0, such that for all x, x + 0 = x.
                            2. Multiplicative
                              1. There is a number, denoted 1, such that for all x, x * 1 = 1 * x = x.
                            3. Inverses
                              1. Additive
                                1. For every x there is a number, denoted -x, such that x + (-x) = 0
                                2. Multiplicative
                                  1. For every nonzero x there is a number, denoted x^−1, such that (x * x^-1) = (x^-1 * x) = 1.
                                3. 0 != 1
                                  1. Irreflexivity of <
                                    1. ~(x < x)
                                    2. Transitivity of <
                                      1. If x < y and y < z, then x < z
                                      2. Trichotomy
                                        1. Either x < y, y < x, or x = y
                                        2. Completeness
                                          1. If a nonempty set of real numbers has an upper bound, then it has a least upper bound.
                                          2. If x < y, then x + y < y + z.
                                            1. If x < y and 0 < z, then x * z < y * z and z * x < z * y.
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