1. Sets, Relations, and Lattices

2. 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

3. 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

4. Venn Diagrams

5. 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)}
Generalization: Ordered n-tuple (a1, a2, …, an)

6. 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)

7. 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

8. 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
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}

9. 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

10. 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 on S “smaller than or equal to” than on S, denoted
if each pair of elements in a common block of is also in a common block of
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}
= {a,b; c,d; e,f; g,h,i}
= {a,f; b,c; d,e; g,h; i}
= {a,b,e,f; c,d; g,h,i}
, but and are incomparable, as are and

11. Totally Ordered Sets
Total ordering: if for every pair a,b in S, either or , 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
Hasse diagram

12. Least/Greatest Member of a Set
Least member: if 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
Greatest member: if 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

13. 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,
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,
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

14. 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: and
Example:
Lattice
Not a lattice

15. 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

16. 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

17. Partially Ordered Set Whose Elements are Partitions
Example: Let = {a,b; c,d,e; f,h; g,i} and = {a,b,c; d,e; f,g; h,i}
= {a,b,c,d,e; f,g,h,i}
= {a,b; c; d,e; f; g; h; i}
= {a,b,c,d,e,f,g,h,i}: greatest partition with just one block
= {a;b;c;d;e;f;g;h;i}: least partition with single-element
blocks

18. 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
= {a;b;c} =
= {a,b;c}
= {a;b,c}
= {a,c;b}
= {a,b,c} =
Product , but , hence lattice not distributive

19. 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:
Distributed and complemented lattice