Advanced Digital Designs Jung H. Kim

Chapter 1. Sets, Relations, and Lattices

Set : a collection of elements from some universe Subset : B  A if any element in B is also in A Proper subset : B  A if B is a subset of A and there is some element in A which is not in B. Operation on sets A  B A  B : all elements in a universe which are not in A. |A| : cardinality - number of distinct elements in A partition  of a set S is a collection of disjoint subset of S whose union is S. ex) A = {a,b,c,d,e,f}  = { }   { } ∵ not disjoint  (  ) : # of elements in the largest block n – tuple (a 1,a 2, …,a n ) : ordered collection on n elements  (a 2, a 1, …,a n ) {a,b,c} = {b,a,c} but (a,b,c)  (b,a,c) Cartesian product of sets A and B is denoted A  B, and contains all ordered pairs A  B = {(a,b) such that a  A and b  B}

Example) A = {a,b,c}, B = {a,e} A  B = {(a,a),(a,e),(b,a),(b,e),(c,a),(c,e)} A set of ordered pairs is a binary relation R. Binary relation R from set A to set B is any subset of A  B. R 1 = {(a,e), (c,e)} R 2  {(a,e), (b,c)} We specify the relation R by the property that relates members of ordered pairs. Properties of R in a set A (a relation from a set A to a set A) 1. R is reflexive iff for all a  A implies (a,a)  R For a set of integer, the relation “  ” is reflexive. 2  I, (2,2)  “  ” ← 2  2For a set of integer, the relation “<” is not reflexive. 2. R is symmetric iff for a,b  A, the existence of the ordered pair (a,b)  R implies that of (b,a)  R. For a set of integer, the relation “  ” is symmetric the relation “=” is not symmetric. 3. R is transitive iff for any three elements a,b,c  A (a,b)  R and (b,c)  R implies (a,c)  R.

For a set of integer, the relation “  ” is not transitive. the relation “<” is transitive. 4. R is antisymmetric iff for any a,b  A, (a,b)  R and a  b implies (b,a)  R that is (a,b)  R and (b,a)  R implies a=b. For a set of integer, the relation “  ” is antisymmetric. the relation “ For a set of integer, R = {(a,b) such that a  |b|} R is not antisymmetric and not symmetric Based on the properties of the binary relation, we can classify into equivalence relation compatibility relation partial ordering relation Equivalence relation is the binary relation which is reflexive, symmetric and transitive. For the integers, R “=” is reflexive, symmetric, and transitive ∴ R(“=”) is equivalence relation Equivalence relation partitions a set into blocks of equivalent elements. R = {(a,b) such that a mod 3 = b mod 3}  reflexive, symmetric and transitive  equivalence relation can be partitioned by = { } : equivalent classes

Compatibility relation is a binary relation which is reflexive, symmetric not necessary transitive, and it can classify elements into non-disjoint called compatibility classes such that all elements in such class are compatible. A = {a,b,c}, R={(a,b),(b,a),(a,c),(c,a),(a,a),(b,b),(c,c)}  reflexive, symmetric not transitive. (b,a)  R, (a,c)  R but (b,c)  R. Compatibility class { } compatibility class and a and c are compatible. Partial ordering relation is a binary relation which is reflexive, antisymmetric, and transitive relation. relation “  ”,  reflexive, antisymmetric and transitive. ∴ R (“  ”) is partial ordering relation. Hasse diagram : graphical representation of P.O. set vertices : elements of the set. Vertex X is a higher level than vertex Y iff X>Y Edge from X to Y is drawn iff X>Y and there is no other element of the P.O set say Z such that X>Z and Z>Y Binary 3-cube (a 1,a 2,a 3 ) (x 1,x 2,x 3 )  (y 1,y 2,y 3 ) iff y i is 1 whenever x i is 1. (0,0,1)  (1,0,1) (1,0,0)  (1,1,0) (1,0,0)  (0,1,1)

(0,0,0) (0,0,1) (0,1,0) (0,1,1) (1,0,0) (1,0,1) (1,1,0) (1,1,1) Binary n cube (a 1,a 2, …,a n ) (x 1,x 2, …,x n )  (y 1,y 2, …,y n ) iff y i is 1 whenever x i = 1. Least upper bound(lub) of a pair of elements a&b of a P.O. set is an element C such that c  a, c  b and there is no other element, say c´ such that c´ < c, c´  a, c´  b. lub need not be unique. lub((0,1,1),(1,1,0)) = (1,1,1) lub((0,1,0),(1,1,0)) = (1,1,0) lub((1,0,0),(0,1,0)) = (1,1,0) Great lower bound(glb) of a and b is an element c such that c  a, c  b and there is no other element c´ such that c´ < c, c´  a, c´  b. glb((0,1,1),(1,1,0))=(0,1,0) glb((0,0,1),(1,0,0))=(0,0,0) glb also need not be unique(generally) A P.O. set for which each pair of elements has a unique glb and lub is called a lattice. We can view glb and lub as binary operations on a lattice. join, +, a + b = lub(a,b)(0,0,1) + (1,0,0) = (1,0,1) meet, ·, a · b = glb(a,b) (0,1,1,) · (1,1,0) = (0,1,0)

Properties of + and · on lattice idempotency : for all elements a ⇒ a + a = a ·a = a commutativity : for a element a and b ⇒ a + b = b + a, a · b = b · a absorptivity : a + (a · b) = a, a · (a + b) = a associativity : a · (b · c) = (a · b) · c, a + (b + c) = (a + b) + c A lattice is distributive iff for any 3 elements a, b, c a · (b + c) = (a · b) + (a · c), a+(b · c) = (a + b) · (a + c) (e.g.) binary 3 – cube  distributive (0,0,1) · ((0,1,0) + (1,0,0)) = (0,0,1) · (1,1,0) = (0,0,0) ((0,0,1) · (0,1,0)) + ((0,0,1) · (1,0,0) = (0,0,0) + (0,0,0) = (0,0,0) bcd e a b · (c + d) = b · e = b ⇒ not distributive (b · c) + (b · d) = a + a = a A lattice is complemented iff for each element a, there exist an element a´ such that a · a´ = 0, a + a´ = 1 (e.g.) binary 3 – cube (0,0,1) · (1,1,0) = 0 (0,0,1) + (1,1,0) = 1 → (0,0,1)´=(1,1,0)

For any (x 1,x 2,x 3 ), the complemented element is (x 1 ´,x 2 ´,x 3 ´) For a, a · e = a, a + e = e → a = e, e´ = a. For element b, b · b´ = a b + b´ = e → b´ = c or d a bcd e Complemented lattice A distributed and complemented lattice is a Boolean algebra. Properties of Boolean algebra. For two well defined operations +, · ① idempotent : a + a = a, a · a = a ② commutative : a + b = b + a, a · b = b · a ③ associatative : a + (b + c) = (a + b) + c, a · (b · c) = (a · b) · c ④ absorptive : a + (a · b) = a, a · (a + b) = a ⑤ distributive : a · (b + c) = (a · b) + (a · c) ⑥ complemented : For any a, there is a unique element a´ such that a · a´ = 0, a + a´ = 1