G4G0-2/Data Structures/Week 1/Workshop 1 - Mathematics Background.md

Lists

• Ordered
• Denoted by (a1, a2, … an) or {a1, a2, …an}

Strings and Queues

Strings

• Ordered list
• (1,2,3,4) or {1,2,3,4}

Queue

• Special list, where elements are removed from the bottom, and added to the top.

Sets, Elements

• Any collection of objects is a set
• Objects contained in a set are elements, or members
• Set defined with braces.
• Conventional to use singular capital letters for names of sets.
• Commonly denoted a a ∈ S, where a is an element of the set S.
  • Elements that are not contained are denoted as d ∉ S, where d is not an element of the set S.

#Cardinality

• Cardinality of a set is the number of elements contained in the set.
• For example, let S = {a,b,c}, the cardinality of S is 3.
• These facts are denoted symbolically
  • n(S) = 3.

Set Equality

• Two sets are equal if they contain exactly the same elements.

#Subsets

• Suppose V is a set, and W is a set formed using only elements of V.
  • W would be a subset of V
• Denoted as W ⊆ V, where W is the subset of V
• Every set is a subset of itself.
  • {Moe, Larry} \subseteq {Moe, Larry}
• The empty set is a subset of every set .
  • {} \subseteq {Moe, Larry}

Example

Let T = {Moe, Larry, Curly}

List all subsets of T.

{Moe} {Larry} {Curly} {Moe, Larry} {Moe, Curly} {Larry, Curly} {Moe, Larry, Curly}

{} <- Empty Set

True or False

1. {b, h, r, q} \subseteq {h, r} - True
2. {a, 13, d, 2} \subseteq {13, 2, d, a} - True