CSE115/ENGR160 Discrete Mathematics
05/03/12
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
s=t
• 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
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
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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-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, ac=(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 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
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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
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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-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
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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
a=b
• ≥ is transitive as if if a≥b and b≥c 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})= {∅, {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
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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 3|9.
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,
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 wellordered as the set of negative integers has no
least element
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