Set Theory Concepts

1
Set Theory Concepts

• Set 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

2
Describing Sets

• Two Ways
• Enumeration list all elements
• 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 lt y lt 14 and is integral

3
Subsets

• When all elements in A are also elements of B
• A is a subset of B
• A ? 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 then
• A 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

4
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 ? B
• A ? B contains the common elements that are in
• both sets A and B
• A ? B x x ? A and x ? B
• The Complement of set A is AC or A
• AC contains all elements in U that are not in A
• A AC U - A
• ACx x ? A and x ?U

5
Properties of Sets
Idempotence Laws A ? A A, A ? A
A Commutative 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) A Distributive Laws A ? (B ? C) (A ? B)
? (A ? C) , A ? (B ? C) (A ? B) ? (A
? C) Involution Law A A Complement Laws
U ?, ? U A ? A U, A ? A
? Identity Laws A ? ? A, A ? U
A A ? U U, A ? ? ? DeMorgans
Laws (A ? B) A ? B, (A ? B) A ? B
6
Venns Diagram
B
A
C
U
7
Difference Operation
B
A
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
8
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 aibi ?
  i where 1 ? i ? n
• The cartesian product or direct product of 2
  sets A and B
• the set of all ordered pairs of A and B
• A ? B
• EXAMPLE
• A0, 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

9
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 y
• x weighs y pounds
• x divides y
• x is the spouse of y
• A Relation, R, is Defined Over
• a set A
• a set B
• a proposition F(x,y)
• R (A, B, F(x,y))
• if F(a,b) is true then aRb
• if F(a,b) is false then aRb

10
Set Relations

• If R ? A ? B , then R is a binary relation
• EXAMPLE R ? A ? B ai ? A bi ? B
• if (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 order
• R -1 (bj,ai)(ai,bj)? R
• R (A, A, F(x,y)) is an equivalence relation on
  set A if
• aRa (reflectivity)
• If aRb then bRa (symmetry)
• If aRb and bRc then aRc (transitivity)
• a, b, c ? A

11
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 true
• Symmetry 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

12
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
• A A1 ? A2 ? ? An
• Either Ai Aj or Ai ? Aj ? (disjoint subsets)
• EXAMPLE
• Consider A1,2,3,,9,10, B11,3, B27,8,10,
  B32,5,6 and B44,9
• A B1 ? B2 ? B3 ? B4
• Bi ? Bj ? ? i ? j
• B1, B2, B3, B4 are Partitions of A

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Equivalence Class

• R is a binary relation over set A
• Partition A into blocks such that
• ax 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 R
• A / R
• The number of equivalence classes rank of R

14
Equivalence Class Example

• R (A, A, F(x,y))
• F(x, y) is Proposition that Kx (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
• a10,3,6,9, a21,4,7,10, a32,5,8
• Each Partition is an Equivalence Class
• A / R 0,3,6,9,1,4,7,10,2,5,8
• Rank of R is 3

15
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 Q
• Q 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

16
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
• A011, 100, 110, 111
• R0(A,A, F0) R1(A, A,F1)
• R0 and R1 are Equivalence Relations
• F0 proposition that all corresponding bits are
  same
• F1 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),(111,0
  11),(111,111)
• R0 is a refinement of R1 R0 ? R1

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Definition of a Function

• A and B are sets, f is a function that maps ai? A
  to bj ? B
• f A ? B
• f(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

18
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)
• EXAMPLE
• B 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)

19
Ordered Relations

• R is a Binary Relation on A
• For a,b,c ? A if the following hold
• aRa (Reflexive Law)
• If aRb and bRa then ab (Anti-Symmetric Law)
• 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

20
Ordered Sets

• R is a binary Relation on A
• For a,b,c? A if the following hold
• aRa (Reflexive Law)
• If aRb and bRa then ab (Anti-Symmetric Law)
• 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 ?

21
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

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Hasse Diagram Examples
(1,1)
1
a
b
(1,0)
(0,1)
c
0
(0,0)
d
(1,1) is the maximal element (0,0) is the minimal
element
1 is the maximal element 0 is the minimal element
f
e
a and b are the maximal elements c is the
greatest lower bound of a, b e and f are the
minimal elements d is the least upper bound of
e, f
23
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 )