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Sets, Relations, and Lattices
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Zvi Kohavi and Niraj K. Jha
Sets
Set: collection of distinct objects
Example: attendees in this class; prime numbers
Objects: elements, or members, of the set
Set with no elements: empty, or null, set
Set of all even numbers between 1 and 10: {2,4,6,8,10}
Infinite set of all positive, even numbers: {2,4,6, …}
Readers of the Kohavi-Jha book living in Antarctica: most likely empty
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Set Definitions
Universe U: set of all possible outcomes
Example: Rolling a die
• U = {f1, f2, f3, f4, f5, f6}
U has 26 = 64 subsets
• Null, {f1}, …, {f6}, {f1,f2}, …, {f5,f6}, {f1,f2,f3}, …, U
A=B: A and B are identical
A  B: A is a subset of B
A  B: A is a proper subset of B
A+B: union of A and B
AB: intersection of A and B
A’: complement of A
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Venn Diagrams
A
A
B
A
A
B
(b) A + B.
(a) AB.
A
(d) AB =
B
(c) A .
B A
.
(e) A
B.
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Ordered Pair
Ordered pair (a,b): specific order associated with a and b
• a: first coordinate
• b: second coordinate
Example: mother and daughter; teacher and student
Example: {(a,a), (a,b), (b,a), (b,c), (c,a)}
a
c
b
Generalization: Ordered n-tuple (a1, a2, …, an)
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Binary Relation
Binary relation R: set of ordered pairs (a,b)
a R b: a is related to b by R
Cartesian product AxB: set of ordered pairs (a,b) s.t. a is in A and b is in B
Example: If A = {p,q} and B = {r,s,t}, then
AxB = {(p,r), (p,s), (p,t), (q,r), (q,s), (q,t)}
Relation from set A to A: relation in A – subset of AxA or A2
Relation R in set A is
• Reflexive if it contains (a,a) for every a in A
• Symmetric if existence of (a,b) in R implies the existence of (b,a)
• Transitive if existence of (b,a) and (a,c) in R implies existence of (b,c)
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Properties of Relations
Relation R in set A is
• Reflexive if it contains (a,a) for every a in A
• Symmetric if existence of (a,b) in R implies the existence of (b,a)
• Transitive if existence of (b,a) and (a,c) in R implies existence of (b,c)
Example: Relation {(a,a), (b,b), (a,b)} – reflexive and transitive, but not
symmetric
Example: Relation {(a,b), (b,a)} – symmetric, but not transitive since it
does not contain (a,a)
Binary relation R in set S: equivalence relation if it is reflexive, symmetric
and transitive
Example: Relation = is an equivalence relation since it satisfies for all
a, b, and c in R
• Reflexive: a = a
• Symmetric: If a = b, then b = a
• Transitive: If a = b and b = c, then a = c
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Equivalence Classes
Equivalence relation: partitions elements of a set into disjoint subsets s.t. all
members of a subset are equivalent and members of different subsets
are not equivalent
Disjoint subsets: equivalence classes
Example: Relation of parallelism between lines in a plane
a
b
e
d
c
f
R = {(a,a), (b,b), (c,c), (d,d), (e,e), (f,f), (a,b), (b,a), (a,c), (c,a), (b,c), (c,b),
(d,e), (e,d)}
Equivalence classes: {a,b,c}, {d,e}, {f}
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Compatibility Relation
Compatibility relation: relation that is reflexive and symmetric, but not transitive
• Nontransitivity  nondisjoint subsets
• Subsets: compatibility classes
Partition: Partition  on set S: collection of disjoint subsets with set union S
• Disjoint subsets: blocks of partition 
• Uniform partition: each block contains the same no. of elements
Example: Equivalence relation for parallel lines induces partition {a,b,c; d,e; f}
Function: set of ordered pairs in which no two pairs have same first coordinate
Example: If A = {a,b,c} and B = {d,e}
• {(a,d), (b,e), (c,d)} is a function from A to B
• {(a,d), (b,e), (c,d), (c,e)} is not
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Partially Ordered Sets
Partial ordering: reflexive, antisymmetric and transitive binary relation
Example: For S = {a,b,c}, partial ordering  satisfies
• Reflexive: a  a
• Symmetric: a  b and b  a imply a=b
• Transitive: if a b and b  c, then a  c
Partition  1on S “smaller than or equal to” than  2 on S, denoted 1   2
• if each pair of elements in a common block of  1is also in a common block of  2
• two partitions incomparable if neither is smaller than or equal to the other
Example: Consider S and its three partitions:
S = {a,b,c,d,e,f,g,h,i}
 1 = {a,b; c,d; e,f; g,h,i}
 2 = {a,f; b,c; d,e; g,h; i}
 3 = {a,b,e,f; c,d; g,h,i}
1   3 , but  1 and  2 are incomparable, as are  2 and  3
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Totally Ordered Sets
Total ordering: if for every pair a,b in S, either a  b or b  a , then S is totally
ordered by binaryrelation
Example: Set of all prime numbers is totally ordered by 
Displaying the ordering relation with a Hasse diagram
Example: Partial ordering displaying divisibility relation among all positive
divisors of 45, such that the quotient is an integer
45
15
9
5
3
1
Hasse diagram
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Least/Greatest Member of a Set
Least member: if a  b for every b in S, then a is called the least member of S
• When least member exists, it is unique
Example: When the set does not have a least member, define minimal
member
a
b
Maximal members
c
d
e
f
Minimal members
Greatest member: if b  a for every b in S, then a is called the greatest
member of S
• When greatest member exists, it is unique
• When greatest member does not exist, define maximal member
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Lower/Upper Bound of a Subset of Set S
Upper bound: Let S be partially ordered and P be a subset of S, then an
element s in S is an upper bound of P if and only if, for every p in P,
ps
• s is not necessarily a member of P
Least upper bound (lub): smallest of all upper bounds
Lower bound: Element s in S is an lower bound of P if and only if, for every p
in P, s  p
Greatest lower bound (glb): largest of all lower bounds
Example: S = {1,3,5,9,15,45} and P = {3,5}
• Upper bounds: 15, 45
• lub: 15
• glb = 1
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Lattice
Lattice: partially ordered set in which every pair of elements has a unique glb
and a unique lub
• Least element: denoted as 0
• Greatest element: denoted as 1
• For each element a of lattice: a  1 and 0  a
Example:
a
b
Maximal members
45
15
c
9
5
d
3
1
e
Lattice
f
Minimal members
Not a lattice
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Lattice (Contd.)
Example: Lattice of all subsets of set S = {a,b,c}, under the ordering relation
of set inclusion, where {a,b,c} = 1 and  = 0
{a,b,c}
{a,b}
{b}
{b,c}
{a}
{a,c}
{c}
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Binary Operation
Because of their uniqueness, lub and glb may be viewed as binary
operations
• Sum a+b = lub(a,b)
• Product a.b = glb(a,b)
lub and glb satisfy:
• Idempotency: a.a = a+a = a
• Commutativity: a.b = b.a and a+b = b+a
• Absorption: a+a.b = a and a.(a+b) = a
• Associativity: a.(b.c) = (a.b).c and a+(b+c) = (a+b)+c
Following properties valid for every finite lattice:
• a+0 = a
• a.0 = 0
• a.1 = a
• a+1 = 1
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Partially Ordered Set Whose Elements
are Partitions
Example: Let  1 = {a,b; c,d,e; f,h; g,i} and  2 = {a,b,c; d,e; f,g; h,i}
 1   2 = {a,b,c,d,e; f,g,h,i}
 1. 2 = {a,b; c; d,e; f; g; h; i}
 (I ) = {a,b,c,d,e,f,g,h,i}: greatest partition with just one block
 (0) = {a;b;c;d;e;f;g;h;i}: least partition with single-element
blocks
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Distributive Law Not Necessarily Valid
Lattice is distributive if and only if
• a.(b+c) = a.b+a.c
• a+(b.c) = (a+b)(a+c)
Example: Consider
 0 = {a;b;c} =  (0)
 1 = {a,b;c}
 2 = {a;b,c}
 3 = {a,c;b}
 4 = {a,b,c} =  (I )
(I)
4=
2
1
0=
3
(0)
Product  1.( 2   3)   1, but  1. 2   1. 3   0, hence lattice not distributive
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Complemented Lattice
Lattice is said to be complemented, if for each element a, there exist a’ s.t.
• a.a’ = 0
• a+a’ = 1
• a’ is the complement of a and vice versa
Example:
{a,b,c}
{a,b}
{b}
{b,c}
{a}
{a,c}
{c}
Distributed and complemented lattice
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