Suppose students in a class are getting together in groups to do an activity.
The teacher wants to make sure of two things:
More precisely, ‘make up all of $\,S\,$’ means that the
union of the subsets is $\,S\,$.
EXAMPLES:
Let
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$\,S=\{1,2,3,4,5\}\,$.
The sets $\,\{1,2,3\}\,$ and $\,\{4,5\}\,$ form a partition of $\,S\,$:
they don't overlap, and, together, they make up all of $\,S\,$.
The sets $\,\{1,2,3\}\,$ and $\,\{4,5\}\,$ form a partition of $\,S\,$:
they don't overlap, and, together, they make up all of $\,S\,$.
The sets
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$\,\{1,5\}\,$ and $\,\{2,3,4\}\,$ form a different partition of $\,S\,$.
The sets
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$\,\{1\}\,$ and $\,\{3\}\,$ and $\,\{2,4,5\}\,$ form a different partition of $\,S\,$.
The sets
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$\,\{1,2\}\,$ and $\,\{3,4\}\,$ do not form a partition of $\,S\,$;
together, they don't make up all of $\,S\,$.
together, they don't make up all of $\,S\,$.
The sets
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$\,\{1,2,3\}\,$ and $\,\{3,4,5\}\,$ do not form a partition of $\,S\,$;
they overlap—the number $\,3\,$ is in more than one of the subsets.
they overlap—the number $\,3\,$ is in more than one of the subsets.
The sets
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$\,\{1,2\}\,$ and $\,\{2,3\}\,$ and $\,\{4,5\}\,$ do not form a partition of $\,S\,$;
they overlap—the number $\,2\,$ is in more than one of the subsets.
they overlap—the number $\,2\,$ is in more than one of the subsets.
BEING ‘RELATED TO EACH OTHER’ IN A SET
One of the most important ways in math to create a partition of a set is by using an equivalence relation,
which is the subject of this section.
Given a set
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$\,S\,$, we first need a way to talk about members of the set being related to
each other.
The symbol ‘$\,\sim\,$’ is
used for ‘is related to’.
The sentence ‘$\,x\sim y\,$’
is read as ‘$\,x\,$ is related to
$\,y\,$’.
The concept is best illustrated with some examples:
- Let
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$\,S\,$ be the set of all people, and let $\,x\,$
and $\,y\,$ be members of $\,S\,$ .
Define: $\,x\sim y\,$ if and only if $\,x\,$ and $\,y\,$ have the same sex
Suppose that Carol and Julia are female; Rick and Karl are male.
TRUE SENTENCE READ AS: WHY TRUE? Carol $\,\sim \,$ Julia ‘Carol is related to Julia’ both Carol and Julia are female Carol $\,\not\sim\,$ Karl ‘Carol is not related to Karl’ Carol and Karl do not have the same sex Rick $\,\sim \,$ Karl ‘Rick is related to Karl’ both Rick and Karl are male -
Let
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$\,S=\{0,1,2,3,\ldots\}\,$.
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$x\sim y\,$ if and only if $\,x\,$ and $\,y\,$ have the same remainder upon division by $\,3\,$
TRUE SENTENCE READ AS: WHY TRUE? [beautiful math coming... please be patient] $\,5\sim 8\,$ ‘$\,5\,$ is related to $\,8\,$’ When $\,5\,$ is divided by $\,3\,$, the remainder is $\,2\,$: $5 = 1\cdot 3 + 2$
When $\,8\,$ is divided by $\,3\,$, the remainder is $\,2\,$: $8 = 2\cdot 3 + 2$
Thus, both have the same remainder ($\,2\,$) when divided by $\,3\,$.$\,3\sim 12\,$ ‘$\,3\,$ is related to $\,12\,$’ When $\,3\,$ is divided by $\,3\,$, the remainder is $\,0\,$: $3 = 1\cdot 3 + 0$
When $\,12\,$ is divided by $\,3\,$, the remainder is $\,0\,$: $12 = 4\cdot 3 + 0$
Thus, both have the same remainder ($\,0\,$) when divided by $\,3\,$.$\,1\not\sim 11\,$ ‘$\,1\,$ is not related to $\,11\,$’ When $\,1\,$ is divided by $\,3\,$, the remainder is $\,1\,$: $1 = 0\cdot 3 + 1$
When $\,11\,$ is divided by $\,3\,$, the remainder is $\,2\,$: $11 = 3\cdot 3 + 2$
Thus, $\,1\,$ and $\,11\,$ have different remainders when divided by $\,3\,$. -
Let
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$\,S\,$ be the set of all ordered pairs of positive integers.
For $\,(a,b)\,$ and $\,(c,d)\,$ in $\,S\,$, define:
[beautiful math coming... please be patient] $\,(a,b) \sim (c,d) \,$ if and only if $\,ad=bc\,$
TRUE SENTENCE READ AS: WHY TRUE? $\,(1,3)\sim (2,6)\,$ ‘$\,(1,3)\,$ is related to $\,(2,6)\,$’ $\,1\cdot 6 = 3\cdot 2\,$ $\,(2,5)\sim (4,10)\,$ ‘$\,(2,5)\,$ is related to $\,(4,10)\,$’ $\,2\cdot 10 = 5\cdot 4\,$ $\,(1,3)\not\sim (2,5)\,$ ‘$\,(1,3)\,$ is not related to $\,(2,5)\,$’ $\,1\cdot 5\neq 3\cdot 2\,$
(This concept is very familiar to you; it's just being cast into an unfamiliar setting.
Hint—think about fractions!)
EQUIVALENCE RELATIONS
Now, we are in a position to define an equivalence relation:
DEFINITION
equivalence relation
Let
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$\,S\,$ be a set.
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$ if and only if
the following properties are satisfied for all members $\,x\,$, $\,y\,$ and $\,z\,$ in $\,S\,$:
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$ if and only if
the following properties are satisfied for all members $\,x\,$, $\,y\,$ and $\,z\,$ in $\,S\,$:
- REFLEXIVE property: $\,x\sim x\,$
(every member is related to itself) - SYMMETRY property: if $\,x\sim y\,$,
then $\,y\sim x\,$
(if one member is related to another, then the other is related to the one) - TRANSITIVE property: if $\,x\sim y\,$ and $\,y\sim z\,$,
then $\,x\sim z\,$
(if one is related to another, and this other is related to a third,
then the first is related to the third)
EXAMPLE: an equivalence relation
on the set of all people
(1) Let
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$\,S\,$ be the set of all people, and let $\,x\,$
and $\,y\,$ be members of $\,S\,$ .
Define: $\,x\sim y\,$ if and only if $\,x\,$ and $\,y\,$ have the same sex
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$, as follows:
Define: $\,x\sim y\,$ if and only if $\,x\,$ and $\,y\,$ have the same sex
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$, as follows:
- reflexivity: every person has the same sex as his/her self
- symmetry: if a person has the same sex as another person, then the other person has the same sex as the first
- transitivity: if person #1 has the same sex as person #2, and person #2 has the same sex as person #3, then person #1 has the same sex as person #3
EXAMPLE: an equivalence relation
on the set of all nonnegative integers
(2) Let
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$\,S=\{0,1,2,3,\ldots\}\,$.
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$\,x\sim y\,$ if and only if $\,x\,$ and $\,y\,$ have the same remainder upon division by $\,3\,$
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$, as follows:
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$\,x\sim y\,$ if and only if $\,x\,$ and $\,y\,$ have the same remainder upon division by $\,3\,$
Then, $\,\sim \,$ is an equivalence relation on $\,S\,$, as follows:
- reflexivity: every number has the same remainder upon division by $\,3\,$ as itself
- symmetry: if a number has the same remainder upon division by $\,3\,$ as another,
then the other has the same remainder upon division by $\,3\,$ as the first - transitivity: if a number has the same remainder upon division by $\,3\,$ as another,
and this other has the same remainder upon division by $\,3\,$ as a third,
then the first number has the same remainder upon division by $\,3\,$ as the third
EXAMPLE: NOT an equivalence relation
(3) Let
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$\,S=\{0,1,2,3,\ldots\}\,$.
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$\,x\sim y\,$ if and only if $\,x\lt y\,$
Then, $\,\sim \,$ is not an equivalence relation on $\,S\,$.
It fails both reflexivity (since $\,x\,$ is not less than $\,x\,$) and symmetry (if $\,x\,$ is less than $\,y\,$, then $\,y\,$ is not less than $\,x\,$).
For $\,x\,$ and $\,y\,$ in $\,S\,$, define:
$\,x\sim y\,$ if and only if $\,x\lt y\,$
Then, $\,\sim \,$ is not an equivalence relation on $\,S\,$.
It fails both reflexivity (since $\,x\,$ is not less than $\,x\,$) and symmetry (if $\,x\,$ is less than $\,y\,$, then $\,y\,$ is not less than $\,x\,$).
EQUIVALENCE CLASSES
Once you select a member of a set which has an equivalence relation on it,
you often want to study all the members which are related to it.
This leads us to:
DEFINITION
equivalence class
Let
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$\,\sim \,$ be an equivalence relation on a set $\,S\,$.
Let $\,x\,$ be a member of $\,S\,$.
The equivalence class determined by $\,x\,$ is the set of all members of $\,S\,$ that are related to $\,x\,$.
Let $\,x\,$ be a member of $\,S\,$.
The equivalence class determined by $\,x\,$ is the set of all members of $\,S\,$ that are related to $\,x\,$.
Here is one key reason why equivalence relations are so important:
The equivalence classes of a set always form a PARTITION of the set.
For Example (1) above, there are only two equivalence classes: males and females.
For Example (2) above, there are three equivalence classes:
-
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$\,\{0,3,6,\ldots\}\,$
(numbers that are divisible by $\,3\,$; that is, the remainder is $\,0\,$) - $\,\{1,4,7,\ldots\}\,$
(numbers that leave a remainder of $\,1\,$ upon division by $\,3\,$) - $\,\{2,5,8,\ldots\}\,$
(numbers that leave a remainder of $\,2\,$ upon division by $\,3\,$)
As a final example,
congruence is an equivalence relation on the set of all geometric figures.
Let
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$\,G_1\,$, $\,G_2\,$, and $\,G_3\,$ be geometric figures. Then:
- Reflexivity: $\,G_1 \cong G_1$ (every geometric figure is congruent to itself)
- Symmetry: if $\,G_1\cong G_2\,$ then $\,G_2\cong G_1\,$
- Transitivity: if $\,G_1\cong G_2\,$ and $\,G_2\cong G_3\,$, then $\,G_1\cong G_3\,$
Master the ideas from this section
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Triangle Congruence
by practicing the exercise at the bottom of this page.
When you're done practicing, move on to:
Triangle Congruence