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: • 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 examples- R on the set A of all people. 5. R={(a,b)| a’s last name starts with the same letter as b’s } R S A T 6. R={(a,b)| a is a (full) sister of b } R S A T 7. R={(a,b)| a has the same major as b } R S A T
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. aRbiff 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, aRbiff a b mod 4 • Recall: Def: a b mod 4 iff ________ • Examples: 7 __ mod 4 • 10 ___ mod 4 1 __ mod 4 • Proof that a b mod 4 is a equivalence relation: • Reflexive:
a b mod 4 • Symmetric: Assume aRb.
Transitive • Transitive: Assume aRb and bRc.
Equivalence classes of a b mod 4 [0] 4 = {
Similarly, do a b mod 3 • Equivalence classes: [0] 3 = {
Ex. 9: R on Reals, aRbiff a-b is an integer • A few examples • Proof this is an equivalence relation: • Reflexive
aRbiff a-b is an integer • Symmetric • Transitive
aRbiff 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)| ab (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??
Proof: • i) ii) 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? Partition? • {(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}
More ex • Relation Equiv Relation? Partition? • {(a,b)|a and b have the same parents} • {(a,b)| a and b share a common parent}
Relation Equiv Relation? Partition? • {(a,b)| a and b speak a common language} • a b mod 3 • Partition: {1,2}, {3}, {4} • Find the relation:
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 • ..
