Download Section 2.1 – Sets As a Basis for Whole Numbers

Chapter 2. Section 1
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Section 2.1 – Sets As a Basis for Whole Numbers
Homework (pages 46-47) problems 1-23
Sets and Their Purpose:
• When working with groups of people or things, it is often useful to organize the data according to
characteristics
• A collection of objects, called elements, is known as a set. The only criteria a set has is that is not
ambiguous – by its definition you know if something is an element of it or not
• For example, are the following examples of sets?
All people whose birthday is October 14
Real numbers
All people who are nice
• For example, say you are comparing houses you want to buy. You are looking for a 3 bedroom, 2
bath home, with a two car garage.
How can this be organized visually?
There are 3 possible sets (number of bedrooms, number of baths and number of garage units)
Do these sets intersect at all (i.e. can you have a house with 3 bedrooms, 2 baths and 2 garage units?)
Yes, all the sets could intersect
So the diagram would look like the following
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Where would a house with 3 bedrooms, 2 baths and 1 car garage go?
This diagram is called a Venn diagram, and it illustrates the relationships between sets. They are
usually circles, and usually have a rectangle around them. The rectangle is known as the Universal
set, which is made up of all possible elements being considered
What would be the universal set in the house example?
Sets can be either finite (a limited number of elements) or infinite (its elements go on forever)
Example, page 47 number 23c. Is the set {0,1,2,3…,200} finite or infinite?
Finite, you could list them all
Chapter 2. Section 1
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Some Terminology:
• Sets are usually denoted by capital letters
• You can define a set in one of three ways
− A description, where you describe what you mean
i.e. A is the set of all days of the week
− A list, which uses curly brackets '{'
i.e. A = {Sunday, Monday, Tuesday, Wednesday, Thursday, Friday, Saturday}
− Set builder notation, which uses a combination of the above methods
i.e. A = { x | x is a day of the week }
• Example, page 46 number 1e. Use the listing method to describe the set of all odd whole numbers
less than 100
{99, 97, 95…11, 9, 7, 5, 3, 1}
• The set with no elements is known as the empty set, sometimes called the null set. It is denoted by
{} or ∅
NOTE: What would the symbol {∅} mean?
• Now because sets are clearly defined, an individual element is either in or out
− Is an element ∈ :
If you want to say that 'Saturday' is a member of the set A above, you would write Saturday ∈ A
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− Is not an element ∉:
If you want to say that 'January' is not a member of the set A above, you would write
January ∉ A
NOTE: With these symbols, you always have an element on the left and a set on the right
Example, page 46 number 2f. True or false {4,3} ⊂ {2,3,4}
True
Relationships Between Sets:
• Let A and B be two sets. A and B are equal (A = B) if and only if they have exactly the same
elements
Formally: for all a ∈ A, then a ∈ B , and for all b ∈ B, then b ∈ A
If A and B are not equal, then we write A ≠ B
NOTE: With these symbols, we have sets on the left and right
• It is standard to not repeat an element in a set, and the ordering of elements does not make a
difference
What is the relationship between A = {1,3,4} and B = {3,4,1}?
• Sets A and B are said to be equivalent ( A : B ) if you can pair the elements of one with the elements
of the other (in other words, they have the same number of elements). Pairing this way is known as
a 1-1 correspondence
• Set A is said to be a subset of a set B ( A ⊆ B ) if and only if every element of A is in B
Formally: for all a ∈ A, then a ∈ B
• Example, page 46 number 3. List all the subsets of {a,b,c}
{a,b,c} {a,b} {a,c} {b,c} {a} {b} {c} { }
Chapter 2. Section 1
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Set A is said to be a proper subset of a set B ( A ⊂ B ) if and only if every element of A is in B and
there is at least one element of B that is not in A. This just eliminates the possibility that A = B
Formally: for all a ∈ A, then a ∈ B and there exists a b ∈ B where b ∉ A
What would the Venn diagram for proper sets look like?
Sets A and B are said to be disjoint if they have no elements in common
Formally: for all a ∈ A, then a ∉ B , and for all b ∈ B, then b ∉ A
What would the Venn diagram for disjoint sets look like?
Joining Sets Together:
• The union of two sets A and B is the set of elements in A or in B
Formally: A ∪ B = {x | x ∈ A or x ∈ B}
• The intersection of sets A and B is the set of all elements common to A and B
Formally: A ∩ B = {x | x ∈ A and x ∈ B}
• The complement of A is the set of all elements in the universe that are not in A
Formally: A = A′ = {x | x ∉ A}
NOTE: There are two different types of notation for the complement
• The difference of set B from set A (relative complement of A with respect to B) is the set of all
elements that are in A and not in B
Formally: A − B = {x | x ∈ A and x ∉ B}
• The Cartesian product of set A with set B is the set of all pairs (a, b)
Formally: A × B = {(a , b) | a ∈ A, b ∈ B}
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Example, page 46 number 6b. Draw the Venn diagram for the set S ∩ T (hint: S ∩ T ≠ ∅ )
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Example, page 46 number 8c. For A = {a,b,c} and B = {b,c} find A – B
A – B = {a}
Example, page 47 number 15. Let A = {3,6,9,12,15,18,21,24…} and B = {6,12,18,24…}
a. Is B ⊆ A ? Yes
b. Find A ∪ B ? Since B ⊆ A , A ∪ B =A
c. Find A ∩ B ? Since B ⊆ A , A ∩ B = B
d. In general, what can you conclude about the above? When B ⊆ A , A ∩ B = B, A ∪ B = A
Example, page 47 number 18d. Find {2,3}× {1,4}
{ (2,1), (2,4), (3,1), (3,4) }
Example, page 47 number 19b. How many elements are in {m, n, o} ×{1,2,3,4}
3 (4) = 12
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