Well-Ordered Set - Discrete Mathematics - Lecture Slides, Slides of Discrete Mathematics

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CSE115/ENGR160 Discrete Mathematics 05/03/

8.4 Equivalence relation

• In traditional C, only the first 8 characters of a variable are checked by the complier
• Let R be relation on the set of strings of characters s.t. sRt where s and t are two strings, if s and t are at least 8 characters long and the first 8 characters of s and t agree, or s=t
• Easy to see R is reflexive, symmetric, and transitive

Equivalence relation

• A relation on a set A is called an equivalence relation if it is reflexive , symmetric , and transitive
• Important property in mathematics and computer science
• Two elements a and b are related by an equivalence relation are called equivalent , denoted by a ∼ b, a and b are equivalent elements w.r.t. a particular equivalence relation (^) Docsity.com^4

Example

• Let R be the relation on the set of integers s.t. aRb iff a=b or a=-b
• We previously showed that R is reflexive, symmetric, and transitive - R is reflexive, aRa iff a=a or a=-a - R is symmetric, aRb if a=b or a=-b, then b=a or b=-a and so bRa (also for the only if part) - R is transitive, if aRb and bRc, then (a=b or a=-b) and (b=c or b=-c). So a=c or a=-c. Thus aRc (also for the only if part)
• If follows that R is an equivalence relation

Example

• Let m be a positive integer with m>1. Show that the relation R={(a,b)|a ≡b (mod m)} is an equivalence relation on the set of integers
• Recall a≡b (mod m) iff m divides a-b. Note that a-a= is divided by m. Hence a≡a (mod m). So congruence modulo m is reflexive
• Suppose a≡b(mod m), then a-b is divisible by m, so a-b=km, where k is an integer. It follows b-a=(-k)m, so b≡a(mod m). Hence, congruence modulo m is symmetric

Example

• Suppose a≡b(mod m) and b≡c(mod m). Then m divides both a-b and b-c. Thus, there are integers k and l with a-b=km and b-c=lm
• Put them together a-c=(a-b)+(b-c) = km+lm = (k+l)m. Thus a≡c(mod m). So, congruence modulo m is transitive
• It follows that congruence modulo m is an equivalence relation

8.6 Partial orderings

• Often use relations to order some or all of the elements of sets
• Example: order words, schedule projects
• A relation R on a set S is called partial ordering or partial order if it is reflexive , antisymmetric , and transitive
• A set S together with a partial ordering R is called partially ordered set , or poset , and is denoted by (S,R)
• Members of S are called elements of the poset

Example

• Show that the “greater than or equal” relation ( ≥) is a partial ordering on the set of integers
• ≥ is reflexive as a ≥ a
• ≥ is antisymmetric as if a ≥ b and b ≥ a then a=b
• ≥ is transitive as if if a≥b and b≥c then a ≥ c
• (Z, ≥) is a poset

Example

• Show that inclusion ⊆ is a partial ordering (the relation of one set being a subset of another is called inclusion) on the power set of a set S
• Example: power set of {0,1,2} is P({0,1,2})= {∅, {0}, {1}, {2}, {0,1}, {0,2}, {1,2}, {0,1,2}}
• A ⊆ A whenever A is a subset of S, so ⊆ is reflexive
• It is antiysmmetric as A ⊆ B and B ⊆ A imply that A=B
• It is transitive as A ⊆ B and B ⊆ C imply that A ⊆ C. Hence ⊆ is a partial ordering on P(S), and (P(S), ⊆) is a poset

Example

• Let R be the relation on the set of people s.t. xRy if x and y are people and x is older than y. Show that R is not a partial ordering
• R is antisymmetric if a person x is order than a person y, then y is not order than x
• R is transitive
• R is not reflexive as no person is older than himself/herself

Comparable

• The elements a and b of a poset (S, ≼) are called comparable if either a≼b or b≼a. When a and b are elements of S s.t. neither a≼b nor b≼a, a and b are incomparable
• In the poset (Z+, |), are the integers 3 and 9 comparable? Are 5 and 7 comparable?
• The integers 3 and 9 are comparable as 3|9. the integers 5 and 7 are incomparable as 5 does not divide 7 and 7 does not divide 5

Total ordering

• Pairs of elements may be incomparable and thus we have “partial” ordering
• When every 2 elements in the set are comparable, the relation is called total ordering
• If (S, ≼) is a poset and every two elements of S are comparable, S is called a totally ordered or linearly ordered set , and ≼ is called a total order or a linear order
• A totally ordered set is also called a chain

Well-ordered set

• (S, ≼) is a well-ordered set if it is a poset s.t. ≼ is a total ordering and every nonempty subset of S has a least element
• The set of ordered pairs of positive integers, Z+×Z +^ with (a 1 , a 2 ) (b 1 , b 2 ) if a 1 < a 2 or if a 1 =b 1 and a 2 ≤b 2 (the lexicographic ordering), is a well-ordered set
• The set Z, with the usual ordering, is not well- ordered as the set of negative integers has no least element (^) Docsity.com^19