SETS

CSC 172

SPRING 2002

LECTURE 20

Sets

Defined by membership relation

Atoms many not have members, but may be members ofa set

Sets may also be members of sets

Membership is “only once” An element can not be a member of a set more than once

Mulitsets, or bags, can contain elements more than once

Sets underlie most information representation

Use of Sets

Sturctured sets represent record structures, tables

Relational models are used in data base systems

Probability space is a kind of set

Set theory underlies much (all?) mathematicsReal numbers, infinities

Representing sets

Extension – list the elements

{1, 2, {3, 4}}

set with 3 elements

Abstraction – describe the elements

{x | 3 <= x <= 10 and x is an integer}

Algebra of Sets

Union : S T = set of elements in S or T or both (+)

Intersection : S T = set of elements in both (*)

Difference: S – T = elements in S, not T (-)

Equality of Sets

Two sets are equal if they have exactly the same members

Two expression involving sets are equivalent () if they produce the same values regardless of what values we assign to the set-variables in the expressions

Algebraic Laws

Observations about pairs of equivalent expressions

Commutative laws of union, intersectionThe order of the operands may be reversed

Associative laws of union, intersectionOperations may be grouped in any order

Similar law for union and difference

S - (T R) (S – T) - R

Algebraic Laws

Distributive lawslike x(y+z) = xy + xz

3 different laws for distribution on sets

S (T R) (S T) (S R)

S (T R) (S T) (S R)

(S T) - R (S - R) (T – R)

Empty Set

is the identity for union S = S

is the annihilator for intersection S = There is no identity for intersection or annihilator for

union because no universal “set containing everything” exists

S – S =

- S = Idempotence : S S = S and S S = S

Proving Equivalences

Three approaches

1. Manipulating known equivalences

2. Classifying elements by sets of which they are members

Venn diagrams and truth-tables

3. Proving containment in both directionsUsing definitions of operations

Manipulating Equivlences

Substituting an expression for all occurrences of some variable

Replacing a subexpression by a known equivalent expression

Using transitivity of equivalenceIf E F and F G then E G

Using commutativity of equivalenceIf E F then F E

Example: (S T) S S

S (T R) (S T) (S R) distributive law

S (T ) (S T) (S ) RS (S T) (S ) Annihilation

S (S T) S Identity

S (S T) S Identity

(S T) S S Commutititvity of

Enumerating Cases

If there are n sets in an expession, we can divide elements into 2n classes, dependin on whether they are in or out of the set

Example: (S T) S S

S T S T RHS

0 0 0 0

0 1 1 0

1 0 1 1

1 1 1 1

Equivalence Through Containment

S T (S is contained in T) means every element of S is an element of T

S = T if and only if (S T) and (T S)

We can prove the equivalence of two expression by showing the result of each is contained in the other

Example: (S T) S S

If x S then x (S T) definition of “union”

If x (S T) then x (S T) S definition of “intersection”

If x (S T) S then x S definition of “intersection”