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Transcript
BASIC SET THEORY
A set can be thought of as
"something that holds things"
More specifically, for every "object" in the
universe, a set S either contains it or it
doesn't. If S contains x we write xS ("x is
in S" or "x is a member of S " ), otherwise
we write x S.
Many sets can be given by listing their
elements. We write out a set given like this
by putting the list inside curly braces.
{ JOHN, PAUL, RINGO, GEORGE}
Sometimes we just show enough to set the
pattern , but this is informal (sometimes even
sloppy) notation:
{1,2,4,8,…,1024} (an 11-element set)
{1,2,4,… }
(an infinite set)
Many more sets can be given by a rule for
determining whether something is a member.
NOBODY UNDER 5 FEET TALL
PERMITTED ON THE ROLLER COASTER
We write out a set given like this by giving a
"dummy" element, a colon, and the test such
an element has to satisfy:
{x: x is at least 5 feet tall}
{x: x is divisible by 2}
these are the
{y: y is divisible by 2}
same set!
The variables x and y above are known as
“dummy variables”; they have no meaning
outside the expression. That’s why we can
change the name.
The question of exactly how a set can be
defined is a very deep one. There is no one
right answer; it's a "choice of rules of play"
like figure skating vs ice dancing or Canadian
football vs American football.
We can leave this one to the mathematicians.
NOTE: An object is either in a set or not in
it. Nothing can be "half in" , "in twice", etc.
{A,B,C} and {A,A,B,C} are the same set.
If set A and set B have the same
elements, then by definition they are two
names for the SAME SET. If either has an
element the other doesn't, they are
DIFFERENT SETS. This is called the
"extensive property" of sets and it is very
important.
In particular, the order does not matter.
{A,B,C} and {C,B,A} are the same set.
A set may have no elements. This is
called an empty set.
THEOREM: There is only one empty set (or,
equivalently, all empty sets are the same.)
Proof: if A is an empty set and B is an
empty set, then they both have no elements,
so they are the same set.
So the set of unicorns and the set of
Sasquatches are (probably) the same set!
We often call this set  , pronounced
"the empty set" rather than "phi".
A set may have one element. To every
object x there corresponds a set {x } called
the singleton set containing x .
THEOREM: {x } = {y } if and only if x = y .
Proof: If {x } = {y }, then:
Any member of {y } is also a member of {x }.
Any member of {x } is equal to x itself.
So any member of {y } is equal to x itself.
So {y } is the singleton set {x }.
The first several weeks of a course in set
theory are filled with annoying theorems like
this that prove very obvious things:
S=S
("reflexivity")
if S=T then T=S
("symmetry")
if R=S and S=T then R=T ("transitivity")
We shall skip most of this!
Sets can be finite or infinite. (A rigorous
definition of "finite set" will have to wait,
but our intuition should be adequate here.)
Every finite set has a cardinality or size.
This is often written as |S| or #(S). For
finite sets this is just the number of
elements.
THEOREM: Two finite sets have the same
size if there is a 1-1 pairing between their
elements.
Fire
Water
Earth
Air
John
Paul (order here is
Ringo arbitrary!!)
George
For infinite sets we adopt this as a definition
of cardinality. The resulting math is very
hard, very interesting, and not part of this
course.
SUBSETS:
A set A is a subset of another set B if
every member of A is also a member of B.
On a Venn diagram:
B
We write AB. (If we know
that some element(s) of B
are NOT in A, we can write AB.)
A
We can interpret “AB” as saying:
“IF x A, THEN x B ”.
THEOREM: If AB and BC then AC.
If AB and BC then AC.
If AB and BC then AC.
INTERSECTION
The intersection of two sets consists of
all the elements that are common to both.
AB = {x: xA and xB}
The membership rule for the set AB is:
The element must satisfy BOTH the rule
for membership of A AND the rule for
membership of B.
EXAMPLES:
{1,2,3}  {2,3,4,5} = {2,3}
{1,3,5,7,9,…}  {1,2,3,…,10} = {1,3,5,7,9}
{1,2,3}  {4,5,6} = 
{Bill, Mary}  {Mary, Jim, Jane} = {Mary}
{A,B,C,D,E}  {D,N,T,O} = {D}
There are lots of obvious, easy, and annoying
theorems about intersection. Many of them
look a lot like addition or multiplication facts.
AA = A
A = 
AB = BA
(AB)C = A(BC)
These can all be proved by going back to the
definition of intersection.
We can visualize "intersection" using a
Venn diagram:
UNION
Another widely used set operation is union.
The union of two sets consists of all the
elements that are in one set, or the other, or
both.
AB = {x: xA or xB}
The membership rule for the set AB is:
The element must satisfy EITHER the
rule for membership of A OR the rule for
membership of B.
Remember, even if an element appears in both
sets, it appears only once in the union.
EXAMPLES:
{1,2,3}  {2,3,4,5} = {1,2,3,4,5}
{1,3,5}  {2,4,6} = {1,2,3,4,5,6}
{1,2,3}  {2} = {1,2,3}
{oats, peas,beans}  {beans, weiners} =?
Unions can be visualized with Venn diagrams.
There are lots of easy theorems about unions,
too:
(AB)  C = A(B  C)
AB = BA
A = A
AA = A
And we also have two distributive laws, so you
don’t have to remember which way it goes:
(AB)C = (AC)  (BC)
(AB)C = (AB)  (AC)
A
B A
C
(fill them in!)
B
C
COMPLEMENTS
If we have a well-defined “universe” U ,
then every set A has a complement A
containing everything that is NOT in A.
The membership rule for the set A is
The element must NOT satisfy the rule
for membership of A.
Venn diagram:
U
A
A
{January, February, March} =
{April,May,...,December}
{1,3,5,...} = ? (U = natural numbers)
{2,4,6,...}
Rules for complements
A=A
U=
=U
AB=AB
AB=AB
“De Morgan’s laws”
“not (A and B) = (not A) or (not B) “
“not (A or B) = (not A) and (not B)”
Don’t drink and drive!
Either don’t drink, or don’t drive.
DIFFERENCE OF SETS
We define the difference of two sets,
A-B, to be the set of all elements that are in
A and not in B. B does not have to be a
subset of A.
A-B = A  B.
EXAMPLES:
{a,b,c} – {a,c} = {b}
{1,2,3,4,5} - {4,5,6,7} = {1,2,3}
THEOREMS: A-A = 
A-  = A
A-B  A
(A-B)  B = 
PRODUCTS
The product AxB of two sets is the set of
ordered pairs (a,b) with aA, bB. Note
that the elements of the product are NOT
elements of the original sets!
{1,2} x {A,B,C} =
{(1,A),(2,A),(1,B),(2,B),(1,C),(2,C)}
{Soup, salad} x {meat,fish,chicken} =
{(soup,meat),(soup,fish),(soup,chicken),
(salad,meat),(salad,fish),(salad,chicken)}
(It’s often good to think about products as
“meals from a menu” where you can take one
choice for each course.)
The membership rule for AxB:
“This is an ordered pair, AND the first
element is in A, AND the second element is in
B.”
We can think of the elements of the product
as arranged in a rectangle:
{a,b,c} x {c,d,e} =
(a,c)
(b,c)
(c,c)
(a,d)
(b,d)
(c,d)
(a,e)
(b,e)
(c,e)
NOTE: the repeated element can be used twice!
If this doesn’t make sense, don’t use a product.
Pick a committee of two out of {Ann, Bill, Cathy}.
As (Ann,Ann) is not a valid committee, we are not
taking a product. [In fact it’s what we call a
“combination”.]
Pick a committee with one man and one woman out
of {Ann, Betty, Cathy},{Paul,Quentin,Rob}.
This is a product; we have 3x3 = 9 choices.
(Ann,Paul)
(Betty,Paul)
(Cathy,Paul)
(Ann,Quentin)
(Betty,Quentin)
(Cathy,Quentin)
Coordinates (x,y) in
the plane consist of one
choice of x and one
choice of y. Thus the
Cartesian plane is RxR,
where R is the real
numbers.
(Ann,Rob)
(Betty,Rob)
(Cathy,Rob)