Transcript Combining Signed Numbers

SETS
Elements of a set
The individual members of the set are called the
elements of the set.
is an element of a set of dishes.
Elements of a set

The symbol means “is an element of”
(hint: looks like an e for element!)
2 1,2,3 means that 2 is an element of the set of
numbers {1, 2, 3}
Elements of a set
Remember Ghostbusters?
4 1,2,3
means 4 is not an element of that set
A well-defined set
A set is “well-defined” if it has clear rules that make
it obvious if something is an element of the set or
not.
A well-defined set?
The set of the members of the 2007 Colts team is a
“well-defined” set.
George Bush – a Colt?
Even though he’s in the picture, you know George Bush is
not really a member of the team because the 2007
Colts team is a “well-defined” set.
NOT a well-defined set
The set of the 5 best Colts players in history is not a
well-defined set because different people may
have different opinions.
Cardinality of a set
The cardinality (or cardinal number) of a set is just
the number of elements in the set.
The cardinality of {2, 4, 6} is 3.
Cardinality of a set
The symbol for the cardinality (or cardinal number)
of a set is n( ).
If A = {2, 4, 6} then n(A) = 3.
Equivalent sets
Two sets are equivalent if they have the same
number of elements or the same cardinality n( ).
If A = {2, 4, 6} and B = {1, 3, 5}
n(A) = 3 and n(B) = 3.
A and B are equivalent sets.
Equal sets
Two sets are equal if they have exactly the same
elements.
Order doesn’t matter.
If A = {2, 4, 6} and B = {6, 4, 2}
A and B are equal sets.
Equal Sets are Equivalent
If A = {2, 4, 6} and B = {6, 4, 2}
A and B are equal sets.
n(A) = 3 and n(B) = 3
A and B are equivalent sets
Equivalent Sets may NOT be Equal
If A = {2, 4, 6} and B = {1, 3, 5}
n(A) = 3 and n(B) = 3
A and B are equivalent sets
but A and B are NOT equal sets.
Naming a set
Sets are traditionally named with capital letters.
Let N = {a, b, c, d, e}
Describing a set
Set are typically enclosed in braces { }
You can describe a set by using:
• words
• a roster
• set-builder notation
Describing a set
The easiest way to describe many sets is by using words.
B = {all blue-eyed blonds in this class}
There isn’t a really good mathematical equivalent for that!
The Roster Method
Just like in gym class when they read the roster,
this simply lists all of the object in the set.
Team A = {Andrews, Baxter, Jones, Smith, Wylie}
When to use a roster
A roster works fine if you only have a few elements
in the set.
However, many sets of numbers are infinite.
Listing each member would take the rest of your life!
When to use a roster
If an infinite set of numbers has a recognizable
pattern, you can still use a roster.
• First establish the pattern
• then use an ellipsis …
to indicate that the pattern goes on indefinitely.
{ 1, 2, 3, 4…}
The Natural Numbers
The set of natural numbers N is also called the set
of counting numbers.
N = { 1, 2, 3, …}
Finite Sets
We think of a finite set as having a “countable”
number of elements.
Mathematically, that means that the cardinality of
the set (number of elements) is a natural number
{1, 2, 3, …}
Geek Patrol: If A is a finite set, n(A)  N
Set-builder notation
For many infinite sets it is easier to just use a rule
to describe the elements of the set.
We use braces { } to let everyone know that we’re
talking about a set.
We use a vertical line | to mean “such that”
let’s see how it works…
Set-builder notation
{x|x>0}
The set of all x such that x is positive.
We couldn’t possible use a roster to describe this
set because it includes fractions and decimals as
well as the counting numbers.
We couldn’t set up the pattern to begin with!
The Empty Set
If a set has NO elements, we call it the empty set.
The symbol that we will use for the empty set is

The cardinal number for the empty set is 0.
OOPS!
Notice that the empty set is the one set that we do not
enclose in braces.
{}
is not empty!
It is a set with one element – the Greek letter Phi.
Vocabulary
Element
Well-defined
Cardinality
Equivalent vs. equal
Finite vs. infinite
Roster vs. set builder notation
Empty set