# 1. Definitions

`// Set A contains elements 1,2 and 3A = {1,2,3}// 2 is an element of A2∈A// 4 is not an element of A4∉A`

# 2. Number Sets

`Naturals    N = {1,2,3,4,...}Integers    Z = {...,-2,-1,0,1,2,...}Rationals   Q = Ratio of 2 integers. example: 1/2Irrationals Q′= Can't be represented as ratios of integersReal        R = Q and Q′Imaginary   I = Everything not in the reals. Ex: (√x = -1)Complex     C = Reals and imaginaries`

# 3. Set Equality

The order and repetition of elements does not matter

`A = {1,2,3}B = {2,3,2,2,3,1,2,3,2,1,1,2,3,2,2,3,1,1,3,2,1}A = B`

# 4. Set Builder Notation

`A = {1,2,3,4,5,6,7,8,9}// B is equal to all elements of A (x∈A), such that (|), // the elements are less than 5 (x<5)B = { x∈A | x<5 }B = {1,2,3,4}`

# 5. Types of Sets

`Universal           U            Empty               {} or ϕFiniteInfiniteSubset`

# 6. Cardinality

The number of distinct elements in a set.

`n(A) or |A|`

# 7. Equivalence

Sets are equivalent when their cardinality is the same. NOT to be mistaken with equality.

`A = {1,2,3,4}B = {5,6,7,8}// A and B are equivalentA~B`

# 8. Subsets

Subsets: ⊆

Proper subsets: ⊂

`A = {1,2,3,4}B = {2,3}// A is a subset of A, but not a PROPER subset of A// ⊆A⊆A// B is a proper subset of AB⊂A`

# 9. Power Sets

A power set P(A) is the set of all of the subsets of A.

`A    = {1,2,3}P(A) = {{},{1},{2},{3},{1,2},{1,3},{2,3},{1,2,3}}`

# 10. Ordered n_tuples / Ordered Pairs

Denoted by round brackets ( ). Unlike sets, the order matters

`(a,b,c) != (c,b,a)// but{a,b,c}  = {c,b,a}`

# 11. Cartesian Products

The cross product between two sets.

`AxB = {(a,b} | a∈A and b∈B}// So if...A   = {1,2,3} and B = {x,y,z}AxB = {(1,x),(1,y),(1,z),(2,x),(2,y),(2,z),(3,x),(3,y),(3,z)}`

The cardinality of A multiplied by the cardinality of B

`n(AxB) = n(A) * n(B)// In our case...n(AxB) = 9`

Think of it as a 2D graph. `(2,1)` is not the same position as `(1,2)`.

` |   x     y     z--------------------1| (1,x) (1,y) (1,z)2| (2,x) (2,y) (2,z)3| (3,x) (3,y) (3,z)`

RxR is the cartesian product of all the real numbers. In other words, every single real number point on an (x,y) graph.

# 12. Operations

How sets interact with each other.

`A∪B = {x | x∈A or x∈B}// So if...A   = {1,2,3} and B = {4,5,6}A∪B = {1,2,3,4,5,6}`
`A∩B = {x | x∈A and x∈B}// So if...A   = {1,2,3} and B = {3,4,5}A∩B = {3}// Another example. If...A = {x∈R | -2<x≤0}B = {x∈R | 0<x<6}A∪B = {x∈R | -2<x<6}A∩B = ϕ`

# 13. Identities 1

Rules that are always true and can help shortcut.

`1. CommutativeA∪B = B∪A2. Associative(A∪B)∪C = A∪(B∪C)3. DistributiveA∪(B∩C) = (A∪B)∩(A∪C)4. EmptyA∪ϕ = AA∩ϕ = ϕ5. UniversalA∪U = UA∩U = A6. SubsetA∪A = AA∩A = A`

# 14. Operations 2

Everything in the universe that isn’t in A

`A  = {x∈R | -2<x<0}Ac = {x∈R | x≤-2 and 0≤x}`

A-B is everything in A that is NOT in B

`A-B = {x | x∈A and x∉B}// So if...A   = {1,2,3,4} and B = {3,4,5}A-B = {1,2}`

# 15. Identities 2

`1. A∪Ac  = U2. (Ac)c = A3. Uc    = ϕ4. ϕc    = U5. A-B   = A∩Bc`

# 16. De Morgans Law

`(A∪B)c = Ac ∩ Bc// And the opposite is true...(A∩B)c = Ac ∪ Bc`

# 17. Partitions

Sets are disjoint when neither have any of the same elements.

`// DisjointA∩B = ϕ`

Mutually Disjoint is where many sets all have no elements in common with each other.

A collection of non-empty sets is a partition if and only if:

1. A is the union of all the sets
2. The sets are mutually disjoint
`// Example 1A = {1,2,3,4,5,6}A₁ = {1,2}A₂ = {3,4,5}A₃ = {6}{A₁,A₂,A₃} is a partition of A because{{1,2},{3,4,5},{6}}Rule 1 and 2 are satisfied// Example 2i = {1,2,3} and Aᵢ = {i,i²} and A= {1,2,3,4,9}Is Aᵢ a partition of A?Aᵢ = {{1},{2,4},{3,9}}YES!`