Download Sets - Lindsay ISD

NOTES on SETS
UNION
Has the following set of events ever occurred to you while shopping? You go into one store and see several
items you would like to buy, and then you go into another store and see more items you would like to buy. You
simply can't decide which object to purchase, you want it all!
In "math talk" what you just experienced is a union.
A union of two or more sets is a third set that contains everything contained in the original two sets. In
language terms, a union can be represented by the logical connector “OR”.
Union is designated by the symbol ∪.
If A and B are sets then A ∪ B represents the union of A and B
In the examples below you'll see examples of the union of sets, then it will be time for you to put together some
sets of your own.
A = {1, 2, 3, 4, 5}, B = {5, 7, 9, 11, 13} then A ∪ B = {1, 2, 3, 4, 5, 7, 9, 11, 13}
Example 1
Notice that when I wrote out the united set I did not write "5" twice. I simply listed all of the new sets elements.
We do not duplicate elements within a set.
A = {all the books written by Charles Dickens}
B = {all the books written by Mark Twain}
A ∪ B = {all books written by either Charles Dickens or Mark Twain}
Example 2
INTERSECTION
Let's go back to our stores for a moment. Have you ever noticed two or more stores that carry the same
merchandise? What you observed is an example of intersection.
The intersection of two (or more) sets is those elements that they have in common. Intersection operation is
designated by the symbol ∩. So if A and B are sets then the intersection (the elements they both have in
common) is denoted by A ∩ B. In language terms, an intersection can be represented by the logical connector
“AND”.
In the following examples you will have a chance to see intersection in action.
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NOTES on SETS
A = {1, 3, 5, 7, 9}, B = {2, 3, 4, 5, 6}.
The elements they have in common are 3 and 5, therefore A ∩ B = {3, 5}.
Example 1
A = {The English alphabet}, B = {vowels}, so A ∩ B = {vowels}
Example 2
A = {1, 2, 3, 4, 5} B = {6, 7, 8, 9, 10}
In this case A and B have nothing in common.
As usual, we have a symbol for this phenomenon: ∅.
∅ is called the “empty set.”
Example 3
Note to reader: Notice that this symbol has no set brackets around it. That is because ∅ literally means a set
with noting in it. ∅ = { }. A ∩ B = ∅
SUBSETS
Let A be the set of objects that you own in your home.
Let B be the set of objects that you own which are kept on the second floor of your home.
Let C be the set of objects that you own which are kept in your bedroom [Note the bedroom is own the second
floor].
Let D be the set of objects that you own which are kept in your bedroom nightstand.
Now we could say D is contained within C, which in turn is contained within B, which in turn is contained
within A.
This is the notion of a subset. When you think about subsets, think about containment.
D is said to be a subset of C since it is completely contained within C (another way to think of this is every
element of set D is also an element contained in set C).
C is said to be a subset of B since it is completely contained within B (another way to think of this is every
element of set C is also an element contained in set A).
B is said to be a subset of A since it is completely contained within A (another way to think of this is every
element of set D is also an element contained in set C).
The symbol for subset is ⊂, so D ⊂ C and C ⊂ B and B ⊂ A.
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NOTES on SETS
However if even one element of one set is not contained within the other then thy are not subsets.
If A were defined as {1,2,3,4,5} and B as {3,4,5,6} then B would not be a subset of A since
“6” ∈ B but 6 ∉ A. The symbol for “not a subset” is ⊄. We would write B ⊄ A.
The notion of subsets is graphically illustrated below
In this first illustration B is entirely within A so B ⊂ A.
In this second illustration A and B have nothing in common (A ∩ B = ∅) so we could write
A ⊄ B and B ⊄ A.
In this last illustration some of B is in A, but not all of B is in A so we could write B ⊄ A.
Finally we’ll look at two more examples, and then I’ll make one final point
A = {1, 2, 3, 4, 5, 6, 7}, B = {2, 3, 4}
B is entirely within A (i.e. every element of B is also an element of A) so we can write B ⊂ A
Example 1
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NOTES on SETS
A = {1, 2, 3, 4, 5} B = {2, 4, 6}, here B ⊄ A
Example 2
One last point to make then you can work out some examples regarding subsets. As in the rest of life
everything is a matter of perspective. “A is a subset of B” could also be viewed as “B contains A.” This notion
of "contain" is symbolized by ⊃. So A ⊂ B is the same thing as B ⊃ A.
SET THEOREMS
First of all a "theorem" is a statement that can be proved.
Below are a few theorems involving the concepts of union, intersection and subsets.
If A ⊂ B and B ⊂ C then A ⊂ C
If A ⊂ B then A ∩ B = A.
If A ⊂ B then A ∪ B = B.
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NOTES on SETS
SETS OF NUMBERS
The sets that we will be concentrating on for the rest of this web site are sets of numbers. Below you will
graphically see the most common sets of numbers, along with brief descriptions of each major set.
Natural numbers are
the numbers you
count on your fingers.
Natural numbers do
not have the number
zero.
Whole numbers are
positive counting
numbers.
Whole numbers do
not have any
decimals and
they cannot be
fractions.
The whole numbers
are {0,1,2,3,4,5,...}
The Real Numbers
Rational means RATIO
Natural
Numbers
Whole Numbers
Definition – Not Rational
Non-repeating, Nonterminating decimals
Integers
Rational Numbers
Rational numbers are any number that
can be expressed as a ratio of two
integers (a ratio being one number
placed over another with a "/" in
between-we will examine this more
thoroughly in the unit on fractions)
{integers} ⊂ {Rational numbers}
The Rational numbers include decimals,
and fractions.
Just as an example of some Rational
numbers, {1/3, 5/1 (–5), –2/3, –2/1 (–2),
0.56, –0.02} ⊂ {Rational numbers}
Irrational Numbers
The Irrational numbers are any number that cannot be
expressed as a ratio of two integers
The Greek constant pi, (π) which is used in
calculating the area and circumference of a circle, is
one example of an Irrational number. π is
approximately equal to 3.14159. Most of the Square
Rootsare Irrational numbers
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Integers are the
positive and
negative counting
numbers.
Integers do not
contain decimals
and
they cannot be
fractions
{Whole
numbers} ⊂
{Integers}
The Integers are
{...–5, –4, –3, –
2, –1 ,0 ,1 ,2 ,3,
4,5,...}
The Real numbers encompass
everything
{Real numbers}={Rational numbers}
∪ {Irrational numbers}.
NOTES on SETS
PROPERTIES OF THE REAL NUMBERS
Okay we arrived at the Real Numbers? So what? The Real Numbers have some very important properties
which I'll explore just for a moment. I know this isn't the most exciting subject but believe it or not these
properties are what makes subjects like algebra work. Just drink a cup of coffee to stay awake, and it will be
over in a minute
•
•
•
•
•
COMMUTATIVE (order of addition or multiplication does not matter)
o The commutative property allows you to perform addition operations and multiplication
operations in any order you wish.
So the commutative property permits a + b = b + a and a × b = b × a.
One example of this is: 3 + 4 = 4 + 3 and 3 × 4 = 4 × 3.
ASSOCIATIVE (how we group the numbers for addition or multiplication does not matter)
o When performing addition or multiplication on three numbers, the associative property allows
you to group them together in any order you wish. (notice the position if the numbers does not
change).
So (a + b) + c = a + (b + c) and (a × b) × c = a × (b × c).
One example of this is: (2 + 3) + 4 = 2 + (3 + 4) and (2 × 3) × 4 = 2 × (3 × 4).
IDENTITY (we get back what we started with)
o the numbers 0 and 1 play an important role in mathematics.
Any number plus 0 equals the original number.
a + 0 = 0 + a = a (you get back what you started with, namely a).
One example of this is: 3 + 0 = 0 + 3 = 3.
0 is called the identity for addition (or additive identity).
Any number multiplied by 1 is equal to the original number.
a × 1 = 1 × a = a (you get back what you started with, namely a).
One example of this is: 3 × 1 = 1 × 3 = 3
1 is called the identity for multiplication (or multiplicative identity).
INVERSE (things that UNDO each other)
o Given any Real Number, a, there exist another Real Number, – a, such that when the two
numbers are added together they equal to zero. These two numbers are called opposites. The
concept of opposites is the reason the Addition Principle of Equality and Inequalities “works” for
equations and inequalities. This means for a Real Number, like 5, there exist another Real
Number, –5, such that when they are added together I get zero. (Example; 5 + (–5) = 0).
o The Inverse property of Real Numbers exist for multiplication. For any Real Number, like 5,
there exist another Real Number, 1/5, such that when they are multiplied together they equal to
one. (Example; 5 × 1/5 = 1). Symbolically, b (1/b) = (1/b) b = 1. b and 1/b are reciprocals of each
other. Then concept of reciprocals is the reason behind the Multiplication Principle of Equality
and Inequalities.
Closure (it means that all the resulting numbers from certain operations are in the same set)
o The closure property means that for any two Real Numbers (pick any two that you would like)
are added or multiplied together then the sum or the product will also be a Real Number. An
example of an operation on Real Numbers that does not result in a Real Number is the solution to
the equation x2 + 1 = 0. result is x = √–1. There is no Real Number such that when it is squared
will equal to –1.
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NOTES on SETS
•
DISTRIBUTIVE
o This one involves addition and multiplication operations being performed together.
It says that a × (b + c) = (a × b) + (a × c) or (b + c) × a = (b × a) + (c × a)
One example of this is: 2 × (3 + 4) = (2 × 3) + (2 × 4)
Set Difference A – B. Everything in set A that is NOT in set B
Set B
Set A
A–B=
Set Complement A’ or NOT A or A. Everything in the Universal set, U, that is NOT in set A. Equivalent to the
set difference U – A
Universe
Set A
Complement of A = NOT A = A =
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