set theory concepts set a collection of elements (objects, members) denoted by upper case letters...

Set Theory ConceptsSet – A collection of “elements” (objects, members)

• denoted by upper case letters A, B, etc.• elements are lower case• brackets are used to encompass members of a set

A = {a, b, c} a A d A

• sets may be finite or infinite is the empty set, = {} is a finite set• U is the universal set, it contains all possible elements• U may be finite or infinite

Describing Sets• Two Ways:

1) Enumeration – list all elements2) Generation – general expression and condition

Example: The set of all integers between 5 and 13

{5,6,7,8,9,10,11,12,13}

{x | 5 x 13 and is integral}

{y | 4 < y < 14 and is integral}

Subsets• When all elements in A are also elements of B:

A is a “subset” of BA B

B “contains” or “covers” A Otherwise, A B Any set is a subset of U is a subset of any set

• If A B and B A, then A = B

• If A B and A B thenA is a “proper subset” of B

A B• The set of subsets of A is the “power set” of A, P(A)

P(A) and A P(A) NOTE: A A and A A and A and A U

Some Common Operations• The “Union” of A and B is A B

A B contains elements that are in set A or in set B or in both sets A and B

A B ={x | x A or x B}

• The “Intersection” of A and B is A BA B contains the common elements that are in

both sets A and BA B ={x | x A and x B}

• The “Complement” of set A is AC or AAC contains all elements in U that are not in A

A = AC = U - AAC={x | x A and x U}

Properties of SetsIdempotence Laws: A A =A, A A = ACommutative Laws: A B = B A, A B = B A Associative Laws: A (B C) = (A B) C,

A (B C) = (A B) C Absorption Laws: A (A B) = A, A (A B)=

ADistributive Laws: A (B C) = (A B) (A

C) , A (B C) = (A B) (A

C) Involution Law: A = AComplement Laws: U = , = U

A A = U, A A = Identity Laws: A = A, A U = A

A U = U, A =

DeMorgan’s Laws: (A B) = A B, (A B) = A B

Venn’s Diagram

A B

CU

Difference Operation

A B

U

A = {1,3,5,6,7,8} B = {1,2,3,4,5}

A – B = {6,7,8}B – A = {2,4}

A B = {1,3,5}

Cartesian Product• 2 elements in a fixed order is a “pair” or “ordered pair”

(a,b)• n elements in a fixed order is an “n-tuple”

(a1, a2, …. , an)(a1, a2, …. , an) = (b1, b2, …. , bn) iff ai=bi i where 1 i n

• The “cartesian product” or “direct product” of 2 sets A and Bthe set of all ordered pairs of A and B

A BEXAMPLE:

A={0, 1} B ={0, 1, 2}A B = {(0,0),(0,1),(1,0),(0,2),(1,1),(1,2)}

• “Cardinality” or “size” of set A is | A |=nA

| A B | = nA nB = 2 3 = 6

Propositional Functions• A Propositional Function, F(x,y), is Defined on A B• Ordered Pair (a,b) Substituted for (x,y) (a,b) A B• F(x,y) Can be a Proposition

(F(x,y) is either true or false, but not both) EXAMPLE:

x is less than yx weighs y poundsx divides yx is the spouse of y

• A Relation, R, is Defined Over:1) a set A2) a set B3) a proposition F(x,y)

R = (A, B, F(x,y))if F(a,b) is true then aRbif F(a,b) is false then aRb

Set Relations• If R A B , then R is a “binary relation”

EXAMPLE: R A B ai A bi Bif (ai,bi) R then ai R bi and “relation R holds”if (ai,bi) R then ai R bi “relation R does not hold”

• Inverse Relation, R -1, is all pairs in R with reverse orderR -1 = {(bj,ai)|(ai,bj) R }

• R =(A, A, F(x,y)) is an “equivalence relation” on set A if:1) aRa

(reflectivity)2) If aRb then bRa (symmetry)3) If aRb and bRc then aRc (transitivity)

a, b, c A

Equivalence Relation• Consider R = (Z, Z, F(x,y)) where Z is the set of all positive

integers and F(x,y) is the Proposition that x = y

R Z Z = {(1,1), (2,2), (3,3) ….}

• For any zi Z it is true that zi R zi

Reflectivity is Satisfied

• For any zi, zj Z, if F(zi,zj) is true then F(zj,zi) is trueSymmetry is Satisfied

• For any zi, zj,zk Z, if F(zi,zj) and F(zj,zk) then F(zj,zk)Transitivity is Satisfied

R is an Equivalence Relation over Z

Set Partitions• A Partition of A denoted by [a] satisfies:

[a] A• Consider a Set of Subsets of A

{A1, A2, …, An}• The Ai are Partitions of A if:

1) A = A1 A2 … An

2) Either Ai = Aj or Ai Aj = (disjoint subsets)

EXAMPLE• Consider A={1,2,3,…,9,10}, B1={1,3}, B2={7,8,10},

B3={2,5,6} and B4={4,9}1) A = B1 B2 B3 B4

2) Bi Bj = i j

{B1, B2, B3, B4} are Partitions of A

Equivalence Class

• R is a “binary relation” over set A• Partition A into “blocks” such that

[a]={x | a R x, x A}• Set [a] is an “equivalence class” of A over R• An arbitrary element of A is a member of exactly one

equivalence class• Set of all equivalence classes over R on A is the “quotient

set” of A wrt RA / R

• The number of equivalence classes “rank” of R

Equivalence Class Example

• R = (A, A, F(x,y))

• F(x, y) is Proposition that K=x (mod 3), K is a Constant

NOTE: F(x, y)= F(x) in this case, a unary proposition

• A ={0,1,2,3,4,5,6,7,8,9,10}

• [a1]={0,3,6,9}, [a2]={1,4,7,10}, [a3]={2,5,8}

• Each Partition is an Equivalence Class

• A / R ={{0,3,6,9},{1,4,7,10},{2,5,8}}

• Rank of R is 3

Logic Notation• “proposition” is a sentence with a clear meaning allowing

its’ evaluation of true or false• Fire is cold - FALSE• Let P and Q be propositions

P Q means that if P holds then Q holds P Q means that P is true iff Q is true, or,

P is a “necessary” and “sufficient” condition for Q

• If P Q :P is a “sufficient condition” of QQ is a “necessary condition” of P

• P Q does not necessarily mean that Q P• Q P is the “converse” of P Q • If P Q then Q P

Q P is the “contraposition” of P Q

Refinement• R1 and R2 are Equivalence Relations over A• if xR1y xR2y for x, y A then

R1 is a “refinement” of R2

R1 R2

EXAMPLE:A={011, 100, 110, 111}

R0=(A,A, F0) R1=(A, A,F1)R0 and R1 are Equivalence Relations

F0 proposition that all corresponding bits are sameF1 is proposition that right two bits are same

R0={(011,011),(100,100),(110,110),(111,111)}R1={(011,011),(011,111),(100,100),(110,110),(111,011),(111,111)}

R0 is a refinement of R1 R0 R1

Definition of a Function• A and B are sets, f is a function that maps ai A to bj B

f: A Bf(ai)=bj

ai f bj

• A is the “domain” of f• bj is the “value” of function f• bj = f(ai)B is an “image” of ai A • A Relation Rf may be Defined from f

f : A B, f(ai)= bj iff (ai, bj) Rf

• f -1 is the “inverse relation” of function f: A B• f -1 is NOT, in general, a function• f -1(bj) IS an “inverse image” of bj

f -1(bj) A

Operation

• “unary” operation is a function, f : A A

• “binary” operation is a function, f : A A A(e.g. ai * aj = ak, (ai,aj) ak)

EXAMPLEB = {0,1} a,b B

a = 1 - a (unary-complement)a b = a • b (binary-conjunction)a b = a + b - (a • b) (binary-disjunction)a b = a + b - (2 • a • b) (binary-exclusive

OR)

Ordered Relations

• R is a Binary Relation on A• For a,b,c A if the following hold:

1) aRa (Reflexive Law)2) If aRb and bRa then a=b (Anti-Symmetric Law)3) If aRb and bRc then aRc (Transitive Law)

• R is said to be a “Partially Ordered Relation”• Also, if a,b A , aRb or bRa then

R is said to be a “Total Order Relation”• Such ordered relations are denoted as

a R b rather than aRb

Ordered Sets

• R is a binary Relation on A• For a,b,c A if the following hold:

1) aRa (Reflexive Law)2) If aRb and bRa then a=b (Anti-Symmetric Law)3) If aRb and bRc then aRc (Transitive Law)

• R is said to be a “Partially Ordered Relation”• Also, if a,b A , aRb or bRa then• R is said to be a “Total Order Relation”• Such ordered relations are denoted as

a R b rather than aRb• An ordered set consists of an order relation and the set over

which it is defined A, R

Hasse Diagrams

• R is a binary Relation on A• For a,b,c A such that a R b and a b

if there is no element c such that a R c, c R b where a b c then b “covers” a

• Hasse Diagrams are useful for visualizing cover characteristics

• Covering elements appear ABOVE Covered elements and are connected by a line

• “Maximal Elements” are those which are NOT Covered

• “Minimal Elements” are those which do NOT cover any other Elements

Hasse Diagram Examples

1 is the maximal element0 is the minimal element

1

0

a b

e f

d

c

(1,1)

(0,0)

(0,1) (1,0)

(1,1) is the maximal element(0,0) is the minimal element

a and b are the maximal elementsc is the greatest lower bound of {a, b}e and f are the minimal elementsd is the least upper bound of {e, f}

Least Upper Bound, Greatest Lower Bound

Let A, R be an ordered set and let B A

• a A is Upper Bound of B if b R a, b B • a A is Lower Bound of B if a R b, b B

• If there is a minimum element in the set of the upper bounds of B, then it is the Least Upper Bound of B (denoted by a b )

• If there is a maximum element in the set of the lower bounds of B, then it is the Greatest Upper Bound of B (denoted by a • b )