Revision of Set Theory

• Defining Sets
• Operations on Sets
• Relations
• Functions
• Summary

Defining Sets: Set Theory

• Sets are unordered collections of elements
• Elements are usually named with lower case letters
• Sets are usually named with capital letters
• Concept of set membership
• Example: `"barina" \in "HOLDENS"`

Defining Sets

• Two basic ways:
  • enumeration
  • comprehension
• Enumeration: `{"barina", "astra", "commodore"}
• Comprehension: `{x | x in S ^^ P}`
  • `S` defines the type of `x`
  • `P` is a predicate in `x`, constraining the set
• Example: `{x | x in NN ^^ x \le 10}`
  • read (the set of natural numbers) x such that x at most 10
  • note a) typing, and b) constraint

Operations on Sets: New Sets from Old

• Union: `S \cup R`
• Intersection: `S \cap R`
• Powerset: `PP(S)`
• Cartesian Product: `S \times R`

See the B summary for formal definitions of these

Relations

• A relation `S harr R` is a formal statement of the correspondence between elements of two sets
• Example: `CARS harr PRICE`
• A relation is a set of sets of ordered pairs:
  `S harr R = PP (S \times R)`
• Note the effect of the powerset: the set of relations `S harr R` includes the empty set (no relation), the set of all ordered pairs `(S \times R)` (everything related to everything), as well as all subsets in between.
• the set of elements in the left of the ordered pairs is called the domain
• the set of elements in the right of the ordered pairs is called the range

Relation Examples

• No price information known: `{}`
• `"barina"` costs $13990: `{("barina",13990)}`
• `"astra"` costs $25490: `{("barina",13990),("astra",25490}`
• "optioned up" `"barina"` costs $25490: `{("barina",13990),("barina",25490),("astra",25490)}`
• Note that elements in both domain and range need not be unique.
• `\dom("BULLET4") = {"barina", "astra"}`
• `\ran("BULLET4") = {13990, 25490}`

Functions

• special case of relations: elements in domain are unique
• not all elements need be in domain: partial
• further special cases:
  • all elements in domain: total
  • elements in range unique: injective
  • all elements in range: surjective
• injective functions are also called one-to-one
• surjective functions are also called onto
• injective/surjective functions can be partial or total
• a function that is both injective and surjective is called bijective

Use of Functions and Relations for Data Modelling

• Since B does not have any data structures in the conventional sense, functions and relations are used
• For example, use function to model an array
  • `array := \{(0,45),(1,23),(2,16),(3,18)\}`
  • `baseprice := \{x\mapsto y | \; x \in HOLDENS ^^ y \in NN\}`
• `x \mapsto y` is called a maplet (ordered pair)
• since the result of a function can itself be a function, can use functions to model classes!

Summary

• Outline of this lecture
  • Defining Sets
  • Operations on Sets
  • Relations
  • Functions
  • Summary
• Sets are an essential component of the B Experience
• understand Relations and Functions as sets
• use Relations and Sets as data structures and class modelling tools

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