CSE115/ENGR160 Discrete Mathematics 05/03/12

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CSE115/ENGR160 Discrete Mathematics05/03/12

• Ming-Hsuan Yang
• UC Merced

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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 st
• Easy to see R is reflexive, symmetric, and
  transitive

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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

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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

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Example

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

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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-a0 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

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Example

• Let m be a positive integer with mgt1. Show that
  the relation R(a,b)a b (mod m) is an
  equivalence relation on the set of integers
• Recall ab (mod m) iff m divides a-b. Note that
  a-a0 is divided by m. Hence aa (mod m). So
  congruence modulo m is reflexive
• Suppose ab(mod m), then a-b is divisible by m,
  so a-bkm, where k is an integer. It follows
  b-a(-k)m, so ba(mod m). Hence, congruence
  modulo m is symmetric

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Example

• Suppose ab(mod m) and bc(mod m). Then m divides
  both a-b and b-c. Thus, there are integers k and
  l with a-bkm and b-clm
• Put them together a-c(a-b)(b-c) kmlm
  (kl)m. Thus ac(mod m). So, congruence modulo m
  is transitive
• It follows that congruence modulo m is an
  equivalence relation

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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-ylt1. Show that R is
  not an equivalence relation
• R is reflexive as x-x0lt1 where x ? R
• R is symmetric for if xRy, then x-ylt1, which
  tells us y-xlt1. So yRx
• R is not transitive. Take x2.8, y1.9, z1.1, so
  that x-y0.9lt1, y-z0.8lt1, but x-z1.7gt1
• So R is not an equivalence relation

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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

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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 ab
• is transitive as if if ab and bc then a c
• (Z, ) is a poset

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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

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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) Ø,
  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
  AB
• 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

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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

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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, ?)

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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 39. the
  integers 5 and 7 are incomparable as 5 does not
  divide 7 and 7 does not divide 5

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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

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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

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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,
  ZZ with (a1, a2) (b1, b2) if a1 lt a2 or if
  a1b1 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