Transcript Slide 1

Pamela Leutwyler
definition
a set is a collection of things –
a set of dishes
a set of clothing
a set of chess pieces
In mathematics, the word “set” refers to
“a well defined collection.”
The “collection of all short people” is not well defined (not a set)
The “collection of all people less than 4 feet tall” is well defined (a set)
notation
a
b
c
d
this collection of letters forms a set
e
{ a , b , c , d , e }
separate the members of the set with commas
surround the members of the set with braces
S= { a , b , c , d , e }
It is conventional to use an upper case letter
to name a set.
S= { a , b , c , d , e }

this symbol means “ is a member of ”
S= { a , b , c , d , e }

this symbol means “ is a member of ”
“d is a member of the set S”
is written
“d  S”
S= { a , b , c , d , e }

this symbol means “ is
not
a member of ”
S= { a , b , c , d , e }

this symbol means “ is
not
a member of ”
“h is not a member of the set S”
is written
“h  S”
S= { a , b , c , d , e }
R= { b , c , e }
S= { a , b , c , d , e }
R= { b , c , e }
the set R is related to the set S
S= { a , b , c , d , e }
R= { b , c , e }
the set R is related to the set S
every element of R is also an element of S
S= { a , b , c , d , e }
R= { b , c , e }
R is called a SUBSET of S
S= { a , b , c , d , e }
R= { b , c , e }
R is called a SUBSET of S
denoted
R

S
the set of CANARIES is
a subset of the set of BIRDS
the set of WOMEN is
a subset of the set of HUMANS
the set of all COUNTING NUMBERS is called
the set of NATURAL NUMBERS and is denoted “N”
N = { 1, 2, 3, 4, 5, 6, 7, 8, … }
the set of all COUNTING NUMBERS is called
the set of NATURAL NUMBERS and is denoted “N”
N = { 1, 2, 3, 4, 5, 6, 7, 8, … }
the set of EVEN counting numbers is a subset of N
EN
two sets are said to be EQUAL if and only if they
contain the same members.
{1, 2, 3 }
= { 2, 1, 3 }
less than OR equal to
less than
in arithmetic we use the symbols:  ,  ,  , 
greater than
greater than OR equal to
If your age is less than 16, you cannot drive legally
a  16
If your age is greater than or equal to 16, you can drive legally
a  16
We use similar notation in set theory.
Suppose we know that every member of a set A
is also a member of a set C.
We use similar notation in set theory.
Suppose we know that every member of a set A
is also a member of a set C.
If it is possible that A is equal to C, we write
AC
example:
A = the set of people enrolled in this class who will
get an A in this class
C = the set of people enrolled in this class
AC
If you know that every member of A
is also a member of C then this
symbol is always correct
BD
If you know that every member of B
is also a member of D and you also know
that there is at least one member of D
that is not in B, you can use this symbol.
In this case, B is called a
PROPER SUBSET of D.
a set that has no members is called “EMPTY” and
is represented with this symbol:

example:
the set of all elephants who can sing opera in French = 
the set that contains everything (or everything that
you are talking about) is called the “UNIVERSE” and
is represented by the symbol:
U
if you are solving algebraic equations then U = the set of real numbers
if you are a doctor studying the correlations of smoking with heart disease then
U = the set of people in your sample population
for any discussion of sets, the universe must be defined.
Sometimes it is difficult or impossible
to list all of the members of a set. We can
use words to describe such sets:
the set of all people
enrolled in this class
the set of all counting numbers
that are greater than 5
the set of all members of the
US senate
“set builder notation” is the most
efficient and accurate way to describe
these sets. A variable symbol is used
to designate an unspecified member of
the Universe:
the set of all people
enrolled in this class
= {x
the set of all counting numbers
that are greater than 5
the set of all members of the
US senate
“set builder notation” is the most
efficient and accurate way to describe
these sets. A variable symbol is used
to designate an unspecified member of
the Universe:
this slash is usually read “such that”
the set of all people
enrolled in this class
= {x/
the set of all counting numbers
that are greater than 5
the set of all members of the
US senate
“set builder notation” is the most
efficient and accurate way to describe
these sets. A variable symbol is used
to designate an unspecified member of
the Universe:
this slash is usually read “such that”
the sentence gives the condition that
defines membership.
the set of all people
enrolled in this class
= { x / x is a person enrolled in this class}
the set of all counting numbers
that are greater than 5
the set of all members of the
US senate
“set builder notation” is the most
efficient and accurate way to describe
these sets. A variable symbol is used
to designate an unspecified member of
the Universe:
this slash is usually read “such that”
the sentence gives the condition that
defines membership.
the set of all people
enrolled in this class
= { x / x is a person enrolled in this class}
the set of all counting numbers
that are greater than 5
= { x / x  N and x  5 }
the set of all members of the
US senate
= { x / x is a US senator }
examples:
{ x / x N and 5  x  9 } = { 6, 7 ,8, 9 }
examples:
{ x / x N and 5  x  9 } = { 6, 7 ,8, 9 }
{ 2x / x N and 5  x  9 } = { 12, 14 ,16, 18 }
the number of members in a set S
is called the “cardinal number of S” denotes “n(S)”
A = { 1, 2, 3, 4 }
V = { a, e, I, o, u }
H = { p, q, r, s }
n(A) = 4
n(V) = 5
n(H) = 4
the number of members in a set S
is called the “cardinal number of S” denotes “n(S)”
A = { 1, 2, 3, 4 }
V = { a, e, I, o, u }
H = { p, q, r, s }
n(A) = 4
Two sets that have the
same cardinal number
are said to be
EQUIVALENT.
n(V) = 5
n(H) = 4
Two sets have the same cardinal number if and only if
it is possible to establish a one to one correspondence
between their members.
E = the set of even counting numbers.
n( E ) = n( N )
E = { 2, 4, 6, 8,……………….. 2n,………}
N = { 1, 2, 3, 4, 5, 6, 7, 8, ……,n,……….}
operations
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
{x/ xA} =
B = { 2, 4, 6, 8, 10 }
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
A’ = { x / x  A } = { 7, 8, 9, 10 }
this set is called the
“complement” of A
and is denoted “ A’ ”
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
{ x / x  A AND x B } =
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
A  B ={ x / x  A AND x B } = { 2, 4, 6 }
this set is called the
“intersection” of A with B
and is denoted
“A  B”
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
{ x / x  A OR x B } =
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1,
1 2,
2 33, 4,
4 5,
5 6}
B = { 2,
2 4,
4 6,
6 8,
8 10 }
{ x / x  A OR x B } =
{
}
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
A  B ={ x / x  A OR x B } =
{ 1, 2, 3, 4, 5, 6, 8, 10 }
this set is called the
“union” of A with B
and is denoted
AB
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
{ x / x  A and x B } =
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
{ x / x  A and x B } = { 1, 3, 5 }
The members of B are removed from A.
This is what remains.
suppose:
U = { 1, 2, 3, 4, 5, 6, 7, 8, 9, 10 }
A = { 1, 2, 3, 4, 5, 6 }
B = { 2, 4, 6, 8, 10 }
A - B = { x / x  A and x B } = { 1, 3, 5 }
this set is called
“A minus B” and is denoted
“A – B”