L27 - Equivalence Classes
CSci/Math 2112
15 July 2015
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Pop Quiz 8
Let R be a relation on a set A. Match the following terms for
properties of R with their definitions:
(i) Transitive
(ii) Reflexive
(iii) Symmetric
(a) For all x, y ∈ A, if xRy , then yRx.
(b) For all x ∈ A, we have xRx.
(c) For all x, y , z ∈ A, if xRy and yRz, then xRz.
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Partitions
The sets A1 , A2 , . . . , Ak are a partition of a set A if
Sk
I
i=1 Ak = A and
I
Ai ∩ Aj = ∅ for all i 6= j.
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Partitions
The sets A1 , A2 , . . . , Ak are a partition of a set A if
Sk
I
i=1 Ak = A and
I
Ai ∩ Aj = ∅ for all i 6= j.
Theorem
Given a relation R on a set A, the equivalence classes of R are a
partition of A.
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Partitions Examples
Example 1
I want to partition the set A = {a, b, c, d} into two sets A1 and A2
(called a bipartition). How many ways are there to do this (order
of the two sets does not matter)?
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Partitions Examples
Example 1
I want to partition the set A = {a, b, c, d} into two sets A1 and A2
(called a bipartition). How many ways are there to do this (order
of the two sets does not matter)?
Example 2
Alice has friends over for a party and wants to play a party game.
For this, she needs to form two groups. Within a group, no two
people are allowed to be friends. Based on the list below of who is
friends with whom, how could Alice accomplish this?
Bob - Celia, Dan, Ella
Celia - Bob, Georgie
Dan - Bob, Frank
Ella - Bob, Georgie
Frank - Dan
Georgie - Celia, Ella
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Relations from Partitions
Theorem
Let A1 , A2 , . . . , Ak be a partition of a set A. Define a relation R on
A by xRy if and only if there is a j such that x, y ∈ Aj . Then R is
an equivalence relation whose equivalence classes are the sets
A1 , . . . , Ak .
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Relations from Partitions
Theorem
Let A1 , A2 , . . . , Ak be a partition of a set A. Define a relation R on
A by xRy if and only if there is a j such that x, y ∈ Aj . Then R is
an equivalence relation whose equivalence classes are the sets
A1 , . . . , Ak .
Example 3
Consider the set A = {2, 3, . . . , 10} with the partition
A1 = {2, 3, 5, 7}, A2 = {4, 6, 8, 9, 10}. What is the relation R on A
based on this partition? How could you write R in set-builder
notation?
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Integers Modulo n
Theorem
When defining addition of integers modulo n as [a] + [b] = [a + b],
there is only one correct solution. (We say that addition is
well-defined.)
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Integers Modulo n
Theorem
When defining addition of integers modulo n as [a] + [b] = [a + b],
there is only one correct solution. (We say that addition is
well-defined.)
Proof (outline).
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Integers Modulo n
Theorem
When defining addition of integers modulo n as [a] + [b] = [a + b],
there is only one correct solution. (We say that addition is
well-defined.)
Proof (outline).
I
Suppose [a] = [a0 ] and [b] = [b 0 ].
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Integers Modulo n
Theorem
When defining addition of integers modulo n as [a] + [b] = [a + b],
there is only one correct solution. (We say that addition is
well-defined.)
Proof (outline).
I
Suppose [a] = [a0 ] and [b] = [b 0 ].
I
Need to show: [a + b] = [a0 + b 0 ]
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Integers Modulo n
Theorem
When defining addition of integers modulo n as [a] + [b] = [a + b],
there is only one correct solution. (We say that addition is
well-defined.)
Proof (outline).
I
Suppose [a] = [a0 ] and [b] = [b 0 ].
I
Need to show: [a + b] = [a0 + b 0 ]
I
Have a − a0 = nx and b − b 0 = ny for some x, y ∈ Z
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Integers Modulo n
Theorem
When defining addition of integers modulo n as [a] + [b] = [a + b],
there is only one correct solution. (We say that addition is
well-defined.)
Proof (outline).
I
Suppose [a] = [a0 ] and [b] = [b 0 ].
I
Need to show: [a + b] = [a0 + b 0 ]
I
Have a − a0 = nx and b − b 0 = ny for some x, y ∈ Z
I
(a + b) − (a0 + b 0 ) = (a − a0 ) + (b − b 0 )
= n(x + y )
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Example 4
Construct the addition and multiplication tables of Z3 , Z4 , and Z6 .
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