Sets, Relations, Maps, Orders and Software
CS 366 Programming Languages

Spring 2015
Robert Muller

The Idea

- Computer software is often required to keep track of "collections"
of things.

- Mathematicians have thought carefully about these collections, and
  know them as sets.

- Software is also often required to keep track of the association
  between items from one set (the "keys") and another (the "values").


Basic Sets 

- A set is a collection of items with no duplicates.

- Examples:
        A = {1, 2, 3}
	B = {Bob, Alice, Joe}
	N = {0, 1, 2, ...}  natural numbers


- NB: Only restriction on elements is identity.


Nomenclature

- Notation:

-- alpha, beta, gamma, Gamma, delta, epsilon

-- A, B, C, ... for sets;
--      or {} the empty set;
-- a, b, c, ... for elements of sets;

-- a in A  means a is an element of set A;

-- (a1, ..., an) is an n-tuple;


Variables and Quantifiers
- x, y, z  for variables (which vary over sets!)

-  forall x in A . statement 
   means statement holds for every element of A;

-  exists x in A . statement 
   means statement holds for some element of A;


Set Comprehensions

- { x | statement } 
    means set of all x such that statement holds;

- Example:
Evens = { x | x in N and exists y in N such that x = 2y }
or
Evens = { x in N | exists y in N such that x = 2y }


Basic Sets

- Notation:

--  Subset : A subseteq B means forall x in A . x in B;

--  Set Equality : A = B means A subseteq B and B subseteq A;


Operations on Sets

- Union : A1 union A2 = { a | a in A1 or a in A2};

- Intersection : A1 intersect A2 = { a | a in A1 and a in A2};

- Disjoint Union : A1 + ... + An = { (a, i) | a in Ai };

- Product : A1 x ... x An = { (a1, ..., an) | ai in Ai };


Operations on Sets : Sequences

- Let A be a set and let e denote empty. 

A* = { w | w = e or w = aw' with a in A & w' in A*}

Example: {a, b}* = {e, ae, ae, aae, abe, ... }

Notes:
- This is an example of an inductive definition,
- Trailing es are usually omitted.


Operations on Sets : Powerset

- Powerset : P(A) = { A' | A' subseteq A};

Example : P({1,2}) = {{}, {1}, {2}, {1, 2}};


Diversion: Russell's Paradox

- Let A = { B | B notin B}.  E.g., {a} in A.

- Assume that A notin A. Then A in A.

- Assume that A in A. Then A notin A.

- So A in A iff A notin A. Naive set theory is inconsistent.


Relations

- R is a(n n-ary) relation on A1, ..., An if and only if R subseteq A1
  x ... x An;

- When R is an n-ary relation on A1, ..., An and A1 = ... = An we say
  that R is an n-ary relation on A1;

- When R is a finite set we say it is a finite relation.


Domain of Definition of a Relation

- For the special case n = 2 of a binary relation R on A1, A2 we define: 

      DomDef(R) = { a1 in A1 | exists a2 in A2 . (a1, a2) in R}


Example Relations

A = {1, 2, 3}, B = {Bob, Alice}

- R1 = A x B = {(1, Bob), (1, Alice), (2, Bob), (2, Alice), (3, Bob),
  (3, Alice)}

- R2 = {}

- R3 = {(1, Bob), (3, Alice)}              // e.g., DomDef(R3) = {1, 3}

- R4 = {(1, Alice), (2, Alice), (3, Alice)}


Partial Maps (aka Partial Functions)

- Let R be a binary relation on A, B. R is a partial map from A to B
  if and only if whenever (a, b) and R and (a, b') and R then b = b'.


Examples

A = {1, 2, 3}, B = {Bob , Alice}

- R1 = A x B = {(1, Bob), (1, Alice), (2, Bob), (2, Alice), (3, Bob),
  (3, Alice)}

- R2 = {}

- R3 = {(1, Bob), (3, Alice)}

- R4 = {(1, Bob), (2, Bob), (3, Bob)}


Maps/Functions

- We usually use f, g, h, ... for partial maps and we write:

             f(a) = b in place of (a, b) in f;

- Let f be a partial map from A to B. f is a total map from A to B iff
  DomDef(f) = A.


Examples

A = {1, 2, 3},   B = {Bob, Alice}

- R1 = A x B = {(1, Bob), (1, Alice), (2, Bob), (2, Alice), (3, Bob), (3, Alice)}

- R2 = {}

- R3 = {(1, Bob), (3, Alice)}

- R4 = {(1, Bob), (2, Bob), (3, Bob)}


New Operations on Sets

- The set of all partial maps from A to B:

  A -o-> B = { f | f is a partial map from A to B }

- The set of all total maps from A to B:

  A --> B = { f | f is a total map from A to B }


Examples

{1, 2} --> {Bob, Alice} = { {(1, Bob), (2, Bob)}, {(1, Alice), (2, Alice)},
                            {(1, Bob), (2, Alice)}, {(1, Alice), (2, Bob)} }

{1, 2} -o-> {Bob, Alice} = {1, 2} --> {Bob , Alice}
                           U { {}, 
                               {(1, Bob)}, {(1, Alice)},
                               {(2, Bob)}, {(2, Alice)}
                           }


Predicates

- Let Bool = {true, false};

  = {void} + {void} = {(void, 0), (void, 1)}

- P is a predicate on A if P is a map from A to Bool.

- Example: >2 is a unary predicate on N:

  >2 = {(0, false), (1, false), (2, false), (3, true), ...} 


Implementing Maps

  Finite maps (from A --> B, or A -o-> B) implemented with a data structure such 
  as hash table, e.g., Java's HashMap

  - The set of "keys" in A must be identifiable (see equivalence below)

  - Other efficient data structures can be used if A is totally ordered


Order

Properties of Relations

- Let R be a relation on A. R is reflexive iff
  forall x in A . (x, x) in R;

- Let R be a relation on A. R is transitive iff
  forall x, y, z in A. If (x,y) in R and (y,z) in R then (x,z) in R

- A relation that is both reflexive and transitive is called a preorder.


Partially Ordered Sets

- Let R be a binary relation on A. R is symmetric iff
  forall x, y in A . If (x, y) in R then (y, x) in R;

- Let R be as above. R is antisymmetric iff
  forall x, y in A. If (x, y) in R and (y, x) in R then x = y;  

- A symmetric preorder is called an equivalence relation.

- An antisymmetric preorder is called a partial order.


Partially Ordered Sets

- If R is a reflexive, antisymmetric and transitive binary relation on A, we say that
  -- R is a partial order on set A
  -- The set A is partially ordered by R
  -- A is a partially ordered set (not mentioning R)
  -- A is a poset


Partially Ordered Sets

- If set A is partially ordered by R, we write 
  -- (A, R) or more often
  -- (A, <=R) or (A, <=) if R is implied in context;

- For a, a' in A, instead of writing (a, a') in R we usually write 
  
  a <=R a' or a <= a' 
  
  if R is implied.

- If a <= a' and a != a' we write a < a'.


Examples

  A = {Bob, Alice},   
  R = {(Bob, Bob), (Alice, Alice), (Bob, Alice)}
                                 
  Hasse Diagram of R

    Alice
      |
     Bob


Examples

A = P({Bob, Alice, Joe}) = { {}, {Bob}, {Alice}, {Joe}, 
                             {Bob, Alice}, {Bob, Joe}, {Alice, Joe},
                             {Bob, Alice, Joe}
                           }

R = subseteq = {({}, {}), ({}, {Bob}), ({}, {Bob, Alice}), ...}

Hasse Diagram of A



Totally Ordered Sets

- Let R be a partial order on A.  R is a total order on A iff
  forall x, y in A . either (x, y) in R or (y, x) in R.

- Example : (N, <=)


Lexicographic Ordering

- Let A be a set (partially ordered by <=A). 

- We derive a partial order <=A* on A* the sequences of elements from A.

- w <=A* w' iff either w = e or
				w = av, w' = a'v' and either
					a <A a' or
					a =A a' and v <=A* v'.


- Example: A = {p, q}, A* = {e, p, q, pq, ppq, ... }

- pq <=A* ppq because a = p, v = q, a' = p, v' = pq, a = a' and v <=A* v' 
  because a = q, v = ?, a' = p, v' = q and v <=A* v' because v = ?.

- If <= is a partial order, then so is <=A*.

In summary: we have type constructors: union, intersection, sum, product, 
sequence, -o-> and -->.

Of these, sum, product, sequence, -o-> have direct computational interpretations.