Discrete Mathematics (c) Marcin Sydow Order relation Discrete Mathematics Quasi-order Divisibility Order Relation Prime numbers GCD and LCM (c) Marcin Sydow Contents Discrete Mathematics (c) Marcin Sydow Order relation Quasi-order partial order relation Divisibility linear order Prime numbers minimal, maximal elements, chains, anti-chains GCD and dense, continuous, well ordering LCM divisibility relation and basic number theory Order relation Discrete Mathematics (c) Marcin 2 Sydow A binary relation R ⊆ X is called a partial order if and only if it is: Order relation Quasi-order 1 reflexive Divisibility 2 anti-symmetric Prime numbers 3 transitive GCD and LCM Denotation: a symbol can be used to denote the symbol of a partial order relation (e.g. a b) Note: a pair (X ; ) where is a partial order on X is also called a poset. “≤” on pairs of numbers? yes aRb , “a divides b” for nonzero integers? yes “<” on pairs of numbers? no ≥ on pairs of numbers yes ⊆ on pairs of subsets of a given universe? yes Examples Discrete Mathematics (c) Marcin Sydow Order relation are the following partial orders?: Quasi-order Divisibility Prime numbers GCD and LCM yes aRb , “a divides b” for nonzero integers? yes “<” on pairs of numbers? no ≥ on pairs of numbers yes ⊆ on pairs of subsets of a given universe? yes Examples Discrete Mathematics (c) Marcin Sydow Order relation are the following partial orders?: Quasi-order Divisibility “≤” on pairs of numbers? Prime numbers GCD and LCM yes “<” on pairs of numbers? no ≥ on pairs of numbers yes ⊆ on pairs of subsets of a given universe? yes Examples Discrete Mathematics (c) Marcin Sydow Order relation are the following partial orders?: Quasi-order Divisibility “≤” on pairs of numbers? yes Prime aRb , “a divides b” for nonzero integers? numbers GCD and LCM no ≥ on pairs of numbers yes ⊆ on pairs of subsets of a given universe? yes Examples Discrete Mathematics (c) Marcin Sydow Order relation are the following partial orders?: Quasi-order Divisibility “≤” on pairs of numbers? yes Prime aRb , “a divides b” for nonzero integers? yes numbers “<” on pairs of numbers? GCD and LCM yes ⊆ on pairs of subsets of a given universe? yes Examples Discrete Mathematics (c) Marcin Sydow Order relation are the following partial orders?: Quasi-order Divisibility “≤” on pairs of numbers? yes Prime aRb , “a divides b” for nonzero integers? yes numbers “<” on pairs of numbers? no GCD and LCM ≥ on pairs of numbers yes Examples Discrete Mathematics (c) Marcin Sydow Order relation are the following partial orders?: Quasi-order Divisibility “≤” on pairs of numbers? yes Prime aRb , “a divides b” for nonzero integers? yes numbers “<” on pairs of numbers? no GCD and LCM ≥ on pairs of numbers yes ⊆ on pairs of subsets of a given universe? Examples Discrete Mathematics (c) Marcin Sydow Order relation are the following partial orders?: Quasi-order Divisibility “≤” on pairs of numbers? yes Prime aRb , “a divides b” for nonzero integers? yes numbers “<” on pairs of numbers? no GCD and LCM ≥ on pairs of numbers yes ⊆ on pairs of subsets of a given universe? yes Comparable and uncomparable elements Discrete Mathematics (c) Marcin Sydow If ⊆ X 2 is a partial order and for some x; y 2 X it holds that Order relation x y or y x we say that elements x; y are comparable in R. Quasi-order Divisibility Prime Otherwise, x and y are uncomparable. numbers GCD and If x y and x 6= y we say x is “smaller” than y or that y is LCM “greater” than x. The word partial reflects that not all pairs of the domain of partial order must be comparable. ≤ on pairs of numbers? yes “a divides b” for non-zero integers? no (show an incomparable pair) ⊆ on pairs of subsets of a given universe? no (show an incomparable pair) Linear order Discrete Mathematics A partial order R that satisfies the following additional 4th (c) Marcin condition: Sydow 8x; y 2 X x y _ y x Order relation (i.e. all elements of the domain are comparable) Quasi-order Divisibility is called linear order. Prime numbers Examples: GCD and which of the following partial orders are linear orders? LCM (in negative cases show at least one pair of incomparable elements) yes “a divides b” for non-zero integers? no (show an incomparable pair) ⊆ on pairs of subsets of a given universe? no (show an incomparable pair) Linear order Discrete Mathematics A partial order R that satisfies the following additional 4th (c) Marcin condition: Sydow 8x; y 2 X x y _ y x Order relation (i.e. all elements of the domain are comparable) Quasi-order Divisibility is called linear order. Prime numbers Examples: GCD and which of the following partial orders are linear orders? LCM (in negative cases show at least one pair of incomparable elements) ≤ on pairs of numbers? no (show an incomparable pair) ⊆ on pairs of subsets of a given universe? no (show an incomparable pair) Linear order Discrete Mathematics A partial order R that satisfies the following additional 4th (c) Marcin condition: Sydow 8x; y 2 X x y _ y x Order relation (i.e. all elements of the domain are comparable) Quasi-order Divisibility is called linear order. Prime numbers Examples: GCD and which of the following partial orders are linear orders? LCM (in negative cases show at least one pair of incomparable elements) ≤ on pairs of numbers? yes “a divides b” for non-zero integers? no (show an incomparable pair) Linear order Discrete Mathematics A partial order R that satisfies the following additional 4th (c) Marcin condition: Sydow 8x; y 2 X x y _ y x Order relation (i.e. all elements of the domain are comparable) Quasi-order Divisibility is called linear order. Prime numbers Examples: GCD and which of the following partial orders are linear orders? LCM (in negative cases show at least one pair of incomparable elements) ≤ on pairs of numbers? yes “a divides b” for non-zero integers? no (show an incomparable pair) ⊆ on pairs of subsets of a given universe? Linear order Discrete Mathematics A partial order R that satisfies the following additional 4th (c) Marcin condition: Sydow 8x; y 2 X x y _ y x Order relation (i.e. all elements of the domain are comparable) Quasi-order Divisibility is called linear order. Prime numbers Examples: GCD and which of the following partial orders are linear orders? LCM (in negative cases show at least one pair of incomparable elements) ≤ on pairs of numbers? yes “a divides b” for non-zero integers? no (show an incomparable pair) ⊆ on pairs of subsets of a given universe? no (show an incomparable pair) Upper and lower bounds Discrete Mathematics (c) Marcin Sydow Order relation If (X ; ) is a poset and A ⊆ X so that for all a 2 A it holds Quasi-order that a u for some u, u is called upper bound of A. Similarly, Divisibility if for all a 2 A it holds that l a, for some l, l is called an Prime numbers lower bound of A. GCD and LCM Example: A = (0; 1) ⊆ R. 5,2,1 are examples of upper bounds of A, -13,-1,0 are examples of lower bounds of A. Maximal and minimal elements Discrete Mathematics (c) Marcin Sydow the element u is maximal element of A ⊆ X , there is no Order element v 6= u in A, so that u v relation Quasi-order the element u is minimal element of A ⊆ X , there is no Divisibility element v 6= u in A, so that v u Prime numbers GCD and Note: there can be more than one maximal or minimal element LCM of a set if they are non-comparable (but there might be no maximal or minimal element of a set) Example: the set (0; 1] ⊆ R has no minimal element. The set of odd naturals has no maximal element. Greatest and Smallest element Discrete Mathematics (c) Marcin Sydow An element is greatest , if it is a unique maximal element and Order it is comparable with all the other elements. relation Quasi-order An element is smallest , if it is a unique minimal element and Divisibility Prime it is comparable with all the other elements. numbers GCD and Note: there could be a unique maximal (minimal) element that LCM is not greatest (smallest), e.g. the poset (Q; ≤) with “artificially” added one element that is not comparable with any other element (it is a unique minimal and maximal but is not greatest nor smallest since it is not comparable with anything) Successor and predecessor Discrete Mathematics (c) Marcin Sydow Order v is a successor of u , v is the minimal of all the relation Quasi-order elements larger than u (denotation: v u) Divisibility v is a predecessor of u , v is the maximal of all the Prime numbers elements smaller than u (denotation: v ≺ u) GCD and LCM Example: in the poset (N; ≤) every element n has a successor (it is n + 1) and every element except 0 has a predecessor. In the poset (Q; ≤) no element has a successor nor predecessor. Chain and antichain Discrete Mathematics (c) Marcin Sydow Let (X ; ) be a poset: Order relation C ⊂ X is called a chain , all pairs of elements of C are Quasi-order comparable Divisibility A ⊂ X is called an anti-chain , all pairs of elements of A Prime numbers are uncomparable GCD and LCM Examples: (f2; 4; 16; 64g; j) is a chain (f3; 5; 8g; j) is an antichain. Hasse diagram Discrete If each non-minimal element has a predecessor and each Mathematics non-maximal element has a successor it is possible to make the (c) Marcin Sydow Hasse Diagram of a poset (X ; ), which is a visualisation of a Order poset.