8.5 Equivalence Relations
Def. of Equivalence Relation
Def: A relation on a set A is called an
equivalence relation if it is R, S, and T
(Reflexive, Symmetric, and Transitive).
Note: These relations split sets into disjoint
classes of equivalent elements where we only
care what class an element is in, not about its
particular identity.
Recall the examples from 8.1
• Ex: Consider the following relations R on the set A of all
people.
Determine which properties (RSAT) hold: circle if so:
1. R={(a,b)| a is older than b }
RSAT
2. R={(a,b)| a lives within 10 miles of b }
RSAT
3. R={(a,b)| a is a cousin of b }
RSAT
4. R={(a,b)| a has the same last name as b }
RSAT
More examplesR on the set A of all people.
5. R={(a,b)| a’s last name starts with the same letter as
b’s }
RSAT
6. R={(a,b)| a is a (full) sister of b }
RSAT
7. R={(a,b)| a has the same major as b }
RSAT
Intro to Equivalence Classes
Note: These relations split sets into disjoint
classes of equivalent elements where we only
care what class an element is in, not about its
particular identity.
Find the equivalence classes on the previous
examples.
More examples, proofs and
equivalence classes
• Ex. 8: Consider the relation R on the set of
integers Z. aRb iff a=b or a= - b.
• Show that R is
– Reflexive
– Symmetric
Ex 8
• Transitive
Equivalence classes
Def: If R is an equivalence relation on a set A,
the equivalence classes of the element a is
[a] R = { s | (a,s) R}.
If b [a] R , b is classed a representative of this
equivalence class.
Details for Ex. 8:
[a] = _________
Ex. 9: R on Z, aRb iff a
b mod 4
• Recall: Def: a b mod 4 iff ________
• Examples: 7 __ mod 4
• 10 ___
mod 4
1 __ mod 4
• Proof that a
relation:
• Reflexive:
b mod 4 is a equivalence
a
b mod 4
• Symmetric: Assume aRb.
Transitive
• Transitive: Assume aRb and bRc.
Equivalence classes of a
[0] 4 = {
b mod 4
Similarly, do a
• Equivalence classes:
[0] 3 = {
b mod 3
Ex. 9: R on Reals,
aRb iff a-b is an integer
• A few examples
• Proof this is an equivalence relation:
• Reflexive
aRb iff a-b is an integer
• Symmetric
• Transitive
aRb iff a-b is an integer
• Some equivalence classes
R={(a,b)| a b (mod m)}
• We’ve considered a few examples (a b mod
3 and a b mod 4). Now, let’s make a general
claim.
• Claim: If m is a positive integers >1, then
R={(a,b)| a b (mod m)} is an equivalence
relation on Z (integers).
• Recall from ch. 2 that a
b mod m iff m|(a-b)
R={(a,b)| a b (mod m)} is an
equivalence relation
• Proof
• Reflexive:
• Symmetric: Assume that aRb.
Transitive:
• Assume that aRb and bRc.
Equivalence classes…
• … of two elements of A are either identical or
disjoint.
Thm. 1
•
•
•
•
•
Let R be an equivalence relation on a set A.
The following are equivalent:
i) aRb
ii) [a] = [b]
iii) [a] [b] ≠
Proof method??
• i)
–
–
–
–
–
–
–
–
–
–
ii)
Proof:
Assume aRb.
Show [a] = [b]. To show =, show ____
To show [a] [b], assume ____ and show ______
Assume c [a]. Then ______
__________________
__________________
__________________
c [b]. So [a] [b]. We’re half done
Other half is similar:
..
..
…proof
• ii)
iii)
– Assume [a] = [b]
– ..
– Therefore [a] [b] ≠
• iii)
i)
– Assume [a] [b] ≠
– ___________
– ___________
– Therefore aRb
Partition
• The equivalence classes form a partition of A (a
collection of disjoint nonempty subsets of A that
have A as their union)
• A=
• Each equivalence relation can be used to
partition the set. Conversely, given a partition
{Ai|i I } of the set S, there is an equivalence
relation R that has the sets Ai as its equivalence
classes.
• Let R= {(x,y)| x and y belong to the same subset
Ai} be the relation.
Recall examples
• Relation
Equiv Relation?
• {(0,0), (1,2), (2,1), (3,3), (1,1), (2,2)} on {1,2,3}
• {(a,a), (b,b), (c,c), (d,d), (a,d), (d,a), (b,c), (c,b)} on {a,b,c,d}
Partition?
More ex
• Relation
Equiv Relation?
•
{(a,b)|a and b have the same parents}
•
{(a,b)| a and b share a common parent}
Partition?
• Relation
Equiv Relation?
• {(a,b)| a and b speak a common language}
• a b mod 3
• Partition: {1,2}, {3}, {4}
• Find the relation:
Partition?
ad=bc
•
•
•
•
Let R be a relation on Z+ x Z+:
((a,b),(c,d)) R iff ad=bc.
Show R is an equivalence relation.
Reflexive
• Symmetric
• Transitive
• ..