Goals
We will
consider what it means for objects to be equal or
indistinguishable,
define equivalence relations (a special type of relation),
practice proving relations are equivalence relations, and
define an important relation on integers.
Which of these things does not belong?
Find any element of each set that does not belong with the others.
4 12 2 8 20 3 , 9 , 6 , 6 , 15
shirts, pants, socks, shoes, bed, belts, sweaters
Which are alike?
Group elements from each set so that the elements of the groups
belong with each other.
4 12 2 8 1 20 7 3 , 9 , 6 , 6 , 3 , 15 , 21
shirts, towels, pants, sheets, pillow cases, sweaters
When are things “alike”?
For each property of relations, determine which are important for
being alike.
1
Reflexive:
2
Symmetric:
3
Anti-Symmetric:
4
Transitive:
Definition
A relation is an equivalence relation iff it is reflexive, symmetric,
and transitive.
Proof
Two points, (a, b) and (c, d) ∈ R2 are related iff ad = bc.
Theorem
The relation R is an equivalence relation.
Proof: R is reflexive. We show that (a, b)R(a, b). Note ab = ab by
reflexivity of equals. Therefore, (a, b)R(a, b) by definition of R.
R is symmetric. Suppose (a, b)R(c, d). Thus ad = bc, by
definition of R. Further cb = da, by symmetry of equals. Finally
(c, d)R(a, b) by definition of R.
R is transitive. Suppose (a, b)R(c, d) and (c, d)R(e, f ). Thus
ad = bc and cf = de by definition of R. By an algebraic
substitution and manipulation ad = b(de/f ), a = be/f , af = be.
Therefore, (a, b)R(e, f ) by definition of R.
Thus R is reflexive, symmetric, and transitive. R is an equivalence
relation by definition of an equivalence relation.
Motivating Example
You are distributing a batch of cookies to your friends. You treat
them fairly by giving each friend the same number of cookies. You
eat only the left over cookies.
1
Choose a number of friends between 4 and 6.
2
For each number from 0 to 20 total cookies, determine how
many cookies you keep.
3
Determine the circumstances under which you keep the most
cookies.
4
Describe these best cases (the total number of cookies) using
mathematical notation.
Modulo Arithmetic
a ≡ b mod n iff n|(b − a).
Is 3 ≡ 5 mod 2?
Is 79 ≡ 33 mod 2?
Is 122 ≡ 343 mod 9?
Practice
a ≡ b mod n iff n|(b − a).
Prove this is an equivalence relation.