CSE115/ENGR160 Discrete Mathematics 05/03/12

1. CSE115/ENGR160 Discrete Mathematics05/03/12 Ming-Hsuan Yang UC Merced

2. 9.4 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

3. Example • The integers a and b are related by the “congruence modulo 4” relation when 4 divides a-b • We will see this relation is reflexive, symmetric and transitive • The above-mentioned two relations are examples of equivalence relations, namely, relations that are reflexive, symmetric, and transitive

4. 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

5. 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 • Ris 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

6. Example • Let R be the relation on the set of real numbers s.t. aRb iff a-b is an integer. Is R an equivalence relation? • Since a-a=0 is an integer for all real number a, aRa for all a. Hence R is reflexive • Now suppose aRb, then a-b is an integer, so b-a is an integer. Hence bRa. Thus R is symmetric • If aRb and bRc, then a-b and b-c are integers. So, a-c=(a-b)+(b-c) is an integer. Hence aRc. Thus R is transitive • Consequently, R is an equivalence relation

7. 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=0 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

8. 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

9. Example • Let R be the relation on the set of real numbers s.t. xRy iff x and y are real numbers that differ by less than 1, that is |x-y|<1. Show that R is not an equivalence relation • R is reflexive as |x-x|=0<1 where x ∊ R • R is symmetric for if xRy, then |x-y|<1, which tells us |y-x|<1. So yRx • R is not transitive. Take x=2.8, y=1.9, z=1.1, so that |x-y|=0.9<1, |y-z|=0.8<1, but |x-z|=1.7>1 • So R is not an equivalence relation

10. 9.6 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

11. 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

12. Example • The divisbility relation | is a partial ordering on the set of positive integers, as it is reflexive, antisymmetric, and transitive • We see that (Z+, |) is a poset

13. 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

14. 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

15. Relation in any poset • In different posets different symbols such as ≤, ⊆ and | are used for a partial ordering • Need a symbol we can use when we discuss the ordering in an arbitrary poset • The notation a ≼ b is used to denote (a,b)∊R in an arbitrary poset (S,R) • ≼ is used as the “less than or equal to” relation on the set of real numbers is the most familiar example of a partial ordering, and similar to ≤ symbol • ≼ is used for any poset, not just “less than or equals” relation , i.e., (S, ≼)

16. 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

17. 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

18. Example • The poset (Z, ≤) is totally ordered as a ≤ b or b ≤ a whenever a and b are integers • The poset (Z+, |) is not tally ordered as it contains elements that are incomparable, such as 5 and 7

19. 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 (a1, a2) (b1, b2) if a1 < a2 or if a1=b1 and a2≤b2 (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