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INTRODUCTION TO SETS
Jump right to the exercises!
The concepts for this exercise are summarized below.
For a complete discussion,
read the text.
(Click here for solutions to the text exercises.)
DEFINITION: set
A set is a collection with the following property:
given any object, either the object
is in the collection, or isn't in the collection. |
"The collection of some people" is not a set; it is too vague.
Is your teacher in this
collection? Maybe or maybe not!
The collection of numbers containing 3, 6, 9, 12, … is a set.
Is the number 35,983,205,119,780,238,482,108,222 in this collection?
Well, either it is (if it's divisible by 3)
or it isn't (if it's not divisible by 3).
Notice that it's not important whether you personally
know whether the answer is YES or NO;
all that matters is that the answer is definitely YES or NO.
NOTES ABOUT SETS:
- A set is a mathematical expression.
- Sets can have different names!
- DEFINITION: The objects in a set are called its elements or its members.
- DEFINITION: If a set has n members, where n
is a whole number, then it is called a finite set (pronounced with a long i, FIGH-night).
- DEFINITION: If a set is not finite, then it is infinite (pronounced with a short i, IN-fi-nit).
EXAMPLES:
{4,10} is a finite set,
with two members.
The number 4 is a member.
The number 10 is a member.
(More on list notation for sets below.)
The set
{1,2
,3,&ldots;}
is an infinite set.
The number 7 is an element of the set.
The number 302
is an element of the set. (The name we use doesn't matter!)
The number 0.25 is not an element of the set.
The following symbols are used in connection with sets:
- { } are called
braces. They are used in list notation for sets (see below).
- ( ) are called
parentheses. (Singular form is parenthesis.) They are used in interval notation for sets (see below).
- [ ] are called
brackets. They are used in interval notation for sets (see below).
LIST NOTATION FOR SETS:
- List notation for sets is used whenever the elements of a set can be listed.
- Braces { } are used for list notation.
- Separate members of the set with commas.
- Use three dots … to indicate that a pattern is to be repeated. Be sure
to list enough elements to clearly establish the pattern.
- The order that elements are listed (for a finite set) doesn't matter.
EXAMPLES:
Here are six names for the same finite set:
{1,2,3}
or
{1,3,2}
or
{2,1,3}
or
{2,3,1}
or
{3,1,2}
or
{3,2,1}
The infinite set
{0,1,2,&ldots;}
contains all the whole numbers.
The finite set
{0,1,2,&ldots;,1000}
contains all the whole numbers between 0 and 1000.
The infinite set
{2,1,0,&ldots;}
contains all the integers that are less than or equal to 2.
These are very different sets!
MEMBERSHIP IN A SET:
- The verb ∈ is used to denote membership
in a set.
- The sentence
x∈S
is read as "ex is in S" or "ex is an element of S" or "ex is a member of S".
- The sentence
x∉
S
is read as "ex is not in S" or "ex is not an element of S" or "ex is not a member of S".
For example, if S is the set
{1,2,3} ,
then all of the following sentences are true:
1∈S
and
2∈S
and
3∈S
and
4∉S
and
84∈S
and
7-4∈S
A SPECIAL SENTENCE:
The sentence "
Let S={1,2,3}
"
is used to assign the name S to the set
{1,2,3}.
The word "let" is the key!
EXAMPLE:
How would a mathematician say "Take the set
{a,b,c,d}
and give it the name T ?
Answer: Let
T={a,b,c,d} .
(The word "let" is a vital part of this sentence!)
DEFINITION: The empty set is the unique set that has no members.
So, the empty set is empty!
The empty set is denoted by ∅
or { } .
Be careful! The set
{∅}
is a set with one member; it is NOT the empty set!
INTERVAL NOTATION FOR SETS:
- Interval notation is used to describe intervals of real numbers.
- Intervals of real numbers are infinite sets.
- Parentheses ( ) are used when an endpoint is not included.
- Brackets [ ] are used when an endpoint is included.
- The "infinity" symbol ∞ is used to denote that an interval
extends forever to the right.
- The "negative infinity" symbol -∞ is used to denote that an interval
extends forever to the left.
- The numbers used in interval notation always go from left to right on the number line.
SUBSETS:
Roughly, a subcollection from a set is called a subset.
For example, let S={a,b} . Then:
{a} is a subset (choose only the "a")
{b} is a subset (choose only the "b")
{a,b} is a subset (choose everything!)
{ } is a subset (choose nothing!)
DEFINITION: subset
Let S be a set.
Set B is called a subset of
S if any one of the following three conditions holds:
- B is the set S itself
- B is the empty set
- each member of B is also a member
of S
|
On this exercise, you will not key in your answer.
However, you can check to see if your answer is correct.