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Set Operations

Description

Senior Freshman Mathematics Mind Map on Set Operations, 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 about 7 years ago
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Resource summary

Set Operations
  1. SUBSET
    1. Basic Subset
      1. A is a subset of B if all elements of A are elements of B.
        1. Proper Subset
          1. A is a proper subset of B if A is a subset of B and A is not equal to B
          2. UNION
              1. A + B
              2. INTERSECTION
                  1. everything the same in A + B
                  2. DISJOINT
                    1. A ∩ B = ∅
                    2. LESS
                      1. A - B (A\B) = A - (A ∩ B)
                      2. COMPLEMENT
                        1. A complement is everything in U outside of A (U\A)
                        2. Commutative
                          1. Associative
                            1. Distributive
                              1. De Morgan Laws
                                1. (A ∩ B) complement = A complement ∪ B complement
                                  1. (A ∪ B)^c = A^c ∩ B^c
                                  2. Involutivity of the Complement
                                    1. (A^c)^c) = A
                                      1. "An involution is a map such that applying it twice gives the (original) identity. Familiar examples: reflecting across the x-axis, the y-axis, or the origin in the plane.
                                      2. Transitivity of Inclusion
                                        1. A ⊆ B ∧ B ⊆ C → A ⊆ C
                                        2. Equality of Sets
                                          1. from (P ↔ Q) ↔ [(P → Q) ∧ (Q → P)]
                                            1. A = B ↔ (A ⊆ B) ∧ (B ⊆ A)
                                              1. Non-Equality of Sets
                                                1. A != B ↔ [(A\B) ∪ (B\A)] != 0
                                              2. POWER SET
                                                1. P(A)
                                                  1. Recall ∅ ⊆ A. Also, A ⊆ A.
                                                    1. If A = {0, 1},
                                                      1. then P(A) = {∅, {0}, {1}, {0,1}}
                                                        1. If A = {a, b, c},
                                                          1. then P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
                                                            1. If A = ∅,
                                                              1. P(A) = {∅}
                                                                1. P(P(A)) = {∅, {∅}}
                                                            2. ∅ and {∅} are different objects. ∅ has no elements, whereas {∅} has one element.
                                                              1. P(A) and A are viewed as living in separate world to avoid phenomena like Russell's paradox.
                                                                1. If A has n elements, then P(A) has 2^n elements.
                                                                  1. In the ZFC (Zermelo Fraenkel set theory) standard system, it is an axion of set theory that every set has a power set, which implies no set consisting of all possible sets could exist, else what would it power set be?

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