# Set Operations

Mind Map by Luke Byrne, updated more than 1 year ago
 Created by Luke Byrne over 2 years ago
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### Description

Senior Freshman Mathematics Mind Map on Set Operations, created by Luke Byrne on 04/22/2018.

## Resource summary

Set Operations
1 SUBSET
1.1 Basic Subset
1.1.1 A is a subset of B if all elements of A are elements of B.
1.1.2 ⊆
1.2 Proper Subset
1.2.1 A is a proper subset of B if A is a subset of B and A is not equal to B
1.2.2 ⊂
2 UNION
2.1 ∪
2.2 A + B
3 INTERSECTION
3.1 ∩
3.2 everything the same in A + B
4 DISJOINT
4.1 A ∩ B = ∅
5 LESS
5.1 A - B (A\B) = A - (A ∩ B)
6 COMPLEMENT
6.1 A complement is everything in U outside of A (U\A)
7 Commutative
8 Associative
9 Distributive
10 De Morgan Laws
10.1 (A ∩ B) complement = A complement ∪ B complement
10.2 (A ∪ B)^c = A^c ∩ B^c
11 Involutivity of the Complement
11.1 (A^c)^c) = A
11.2 "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.
12 Transitivity of Inclusion
12.1 A ⊆ B ∧ B ⊆ C → A ⊆ C
13 Equality of Sets
13.1 from (P ↔ Q) ↔ [(P → Q) ∧ (Q → P)]
13.2 A = B ↔ (A ⊆ B) ∧ (B ⊆ A)
13.3 Non-Equality of Sets
13.3.1 A != B ↔ [(A\B) ∪ (B\A)] != 0
14 POWER SET
14.1 P(A)
14.1.1 Recall ∅ ⊆ A. Also, A ⊆ A.
14.1.2 If A = {0, 1},
14.1.2.1 then P(A) = {∅, {0}, {1}, {0,1}}
14.1.2.2 If A = {a, b, c},
14.1.2.2.1 then P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
14.1.2.2.2 If A = ∅,
14.1.2.2.2.1 P(A) = {∅}
14.1.2.2.2.2 P(P(A)) = {∅, {∅}}
14.1.3 ∅ and {∅} are different objects. ∅ has no elements, whereas {∅} has one element.
14.1.4 P(A) and A are viewed as living in separate world to avoid phenomena like Russell's paradox.
14.1.5 If A has n elements, then P(A) has 2^n elements.
14.1.6 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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