13397320
Description
Mind Map by Luke Byrne, updated more than 1 year ago
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Created by Luke Byrne
about 7 years ago
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Set Operations
- SUBSET
- Basic Subset
- A is a subset of B if all elements of A are elements of B.
- ⊆
- A is a subset of B if all elements of A are elements of B.
- Proper Subset
- A is a proper subset of B if A is a subset of B and A is not equal to B
- ⊂
- A is a proper subset of B if A is a subset of B and A is not equal to B
- Basic Subset
- UNION
- ∪
- A + B
- ∪
- INTERSECTION
- ∩
- everything the same in A + B
- ∩
- DISJOINT
- A ∩ B = ∅
- A ∩ B = ∅
- LESS
- A - B (A\B) = A - (A ∩ B)
- A - B (A\B) = A - (A ∩ B)
- COMPLEMENT
- A complement is everything in U outside of A (U\A)
- A complement is everything in U outside of A (U\A)
- Commutative
- Associative
- Distributive
- De Morgan Laws
- (A ∩ B) complement
= A complement ∪ B
complement
- (A ∪ B)^c = A^c ∩ B^c
- (A ∩ B) complement
= A complement ∪ B
complement
- Involutivity of the Complement
- (A^c)^c) = A
- "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.
- (A^c)^c) = A
- Transitivity of Inclusion
- A ⊆ B ∧ B ⊆ C → A ⊆ C
- A ⊆ B ∧ B ⊆ C → A ⊆ C
- Equality of Sets
- from (P ↔ Q) ↔ [(P → Q) ∧ (Q → P)]
- A = B ↔ (A ⊆ B) ∧ (B ⊆ A)
- Non-Equality of Sets
- A != B ↔ [(A\B) ∪ (B\A)] != 0
- A != B ↔ [(A\B) ∪ (B\A)] != 0
- from (P ↔ Q) ↔ [(P → Q) ∧ (Q → P)]
- POWER SET
- P(A)
- Recall ∅ ⊆ A. Also, A ⊆ A.
- If A = {0, 1},
- then P(A) = {∅, {0}, {1}, {0,1}}
- If A = {a, b, c},
- then P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
- If A = ∅,
- P(A) = {∅}
- P(P(A)) = {∅, {∅}}
- P(A) = {∅}
- then P(A) = {∅, {a}, {b}, {c}, {a, b}, {a, c}, {b, c}, {a, b, c}}.
- then P(A) = {∅, {0}, {1}, {0,1}}
- ∅ and {∅} are different objects. ∅ has no elements, whereas {∅} has one element.
- P(A) and A are viewed as living in separate world to avoid phenomena like Russell's paradox.
- If A has n elements, then P(A) has 2^n elements.
- 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?
- Recall ∅ ⊆ A. Also, A ⊆ A.
- P(A)
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