Презентация Discrete mathematics. Sets онлайн
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Слайды и текст к этой презентации:
№2 слайд
Содержание слайда: What is a set?
A set is a group of “objects”
People in a class: { Alice, Bob, Chris }
Classes offered by a department: { CS 101, CS 202, … }
Colors of a rainbow: { red, orange, yellow, green, blue, purple }
States of matter { solid, liquid, gas, plasma }
States in the US: { Alabama, Alaska, Virginia, … }
Sets can contain non-related elements: { 3, a, red, Virginia }
Although a set can contain (almost) anything, we will most often use sets of numbers
All positive numbers less than or equal to 5: {1, 2, 3, 4, 5}
A few selected real numbers: { 2.1, π, 0, -6.32, e }
№4 слайд
Содержание слайда: Set properties 2
Sets do not have duplicate elements
Consider the set of vowels in the alphabet.
It makes no sense to list them as {a, a, a, e, i, o, o, o, o, o, u}
What we really want is just {a, e, i, o, u}
Consider the list of students in this class
Again, it does not make sense to list somebody twice
Note that a list is like a set, but order does matter and duplicate elements are allowed
We won’t be studying lists much in this class
№5 слайд
Содержание слайда: Specifying a set 1
Sets are usually represented by a capital letter (A, B, S, etc.)
Elements are usually represented by an italic lower-case letter (a, x, y, etc.)
Easiest way to specify a set is to list all the elements: A = {1, 2, 3, 4, 5}
Not always feasible for large or infinite sets
№6 слайд
Содержание слайда: Specifying a set 2
Can use an ellipsis (…): B = {0, 1, 2, 3, …}
Can cause confusion. Consider the set C = {3, 5, 7, …}. What comes next?
If the set is all odd integers greater than 2, it is 9
If the set is all prime numbers greater than 2, it is 11
Can use set-builder notation
D = {x | x is prime and x > 2}
E = {x | x is odd and x > 2}
The vertical bar means “such that”
Thus, set D is read (in English) as: “all elements x such that x is prime and x is greater than 2”
№7 слайд
Содержание слайда: Specifying a set 3
A set is said to “contain” the various “members” or “elements” that make up the set
If an element a is a member of (or an element of) a set S, we use then notation a S
4 {1, 2, 3, 4}
If an element is not a member of (or an element of) a set S, we use the notation a S
7 {1, 2, 3, 4}
Virginia {1, 2, 3, 4}
№8 слайд
Содержание слайда: Often used sets
N = {0, 1, 2, 3, …} is the set of natural numbers
Z = {…, -2, -1, 0, 1, 2, …} is the set of integers
Z+ = {1, 2, 3, …} is the set of positive integers (a.k.a whole numbers)
Note that people disagree on the exact definitions of whole numbers and natural numbers
Q = {p/q | p Z, q Z, q ≠ 0} is the set of rational numbers
Any number that can be expressed as a fraction of two integers (where the bottom one is not zero)
R is the set of real numbers
№9 слайд
Содержание слайда: The universal set 1
U is the universal set – the set of all of elements (or the “universe”) from which given any set is drawn
For the set {-2, 0.4, 2}, U would be the real numbers
For the set {0, 1, 2}, U could be the natural numbers (zero and up), the integers, the rational numbers, or the real numbers, depending on the context
№10 слайд
Содержание слайда: The universal set 2
For the set of the students in this class, U would be all the students in the University (or perhaps all the people in the world)
For the set of the vowels of the alphabet, U would be all the letters of the alphabet
To differentiate U from U (which is a set operation), the universal set is written in a different font (and in bold and italics)
№13 слайд
Содержание слайда: The empty set 1
If a set has zero elements, it is called the empty (or null) set
Written using the symbol
Thus, = { } VERY IMPORTANT
If you get confused about the empty set in a problem, try replacing by { }
As the empty set is a set, it can be a element of other sets
{ , 1, 2, 3, x } is a valid set
№15 слайд
Содержание слайда: Set equality
Two sets are equal if they have the same elements
{1, 2, 3, 4, 5} = {5, 4, 3, 2, 1}
Remember that order does not matter!
{1, 2, 3, 2, 4, 3, 2, 1} = {4, 3, 2, 1}
Remember that duplicate elements do not matter!
Two sets are not equal if they do not have the same elements
{1, 2, 3, 4, 5} ≠ {1, 2, 3, 4}
№16 слайд
Содержание слайда: Subsets 1
If all the elements of a set S are also elements of a set T, then S is a subset of T
For example, if S = {2, 4, 6} and T = {1, 2, 3, 4, 5, 6, 7}, then S is a subset of T
This is specified by S T
Or by {2, 4, 6} {1, 2, 3, 4, 5, 6, 7}
If S is not a subset of T, it is written as such:
S T
For example, {1, 2, 8} {1, 2, 3, 4, 5, 6, 7}
№18 слайд
Содержание слайда: Subsets 3
The empty set is a subset of all sets (including itself!)
Recall that all sets are subsets of themselves
All sets are subsets of the universal set
A horrible way to define a subset:
x ( xA xB )
English translation: for all possible values of x, (meaning for all possible elements of a set), if x is an element of A, then x is an element of B
This type of notation will be gone over later
№19 слайд
Содержание слайда: Proper Subsets 1
If S is a subset of T, and S is not equal to T, then S is a proper subset of T
Let T = {0, 1, 2, 3, 4, 5}
If S = {1, 2, 3}, S is not equal to T, and S is a subset of T
A proper subset is written as S T
Let R = {0, 1, 2, 3, 4, 5}. R is equal to T, and thus is a subset (but not a proper subset) or T
Can be written as: R T and R T (or just R = T)
Let Q = {4, 5, 6}. Q is neither a subset or T nor a proper subset of T
№22 слайд
Содержание слайда: Set cardinality
The cardinality of a set is the number of elements in a set
Written as |A|
Examples
Let R = {1, 2, 3, 4, 5}. Then |R| = 5
|| = 0
Let S = {, {a}, {b}, {a, b}}. Then |S| = 4
This is the same notation used for vector length in geometry
A set with one element is sometimes called a singleton set
№23 слайд
Содержание слайда: Power sets 1
Given the set S = {0, 1}. What are all the possible subsets of S?
They are: (as it is a subset of all sets), {0}, {1}, and {0, 1}
The power set of S (written as P(S)) is the set of all the subsets of S
P(S) = { , {0}, {1}, {0,1} }
Note that |S| = 2 and |P(S)| = 4
№25 слайд
Содержание слайда: Tuples
In 2-dimensional space, it is a (x, y) pair of numbers to specify a location
In 3-dimensional (1,2,3) is not the same as (3,2,1) – space, it is a (x, y, z) triple of numbers
In n-dimensional space, it is a
n-tuple of numbers
Two-dimensional space uses
pairs, or 2-tuples
Three-dimensional space uses
triples, or 3-tuples
Note that these tuples are
ordered, unlike sets
the x value has to come first
№26 слайд
Содержание слайда: Cartesian products 1
A Cartesian product is a set of all ordered 2-tuples where each “part” is from a given set
Denoted by A x B, and uses parenthesis (not curly brackets)
For example, 2-D Cartesian coordinates are the set of all ordered pairs Z x Z
Recall Z is the set of all integers
This is all the possible coordinates in 2-D space
Example: Given A = { a, b } and B = { 0, 1 }, what is their Cartiesian product?
C = A x B = { (a,0), (a,1), (b,0), (b,1) }
№28 слайд
Содержание слайда: Cartesian products 3
All the possible grades in this class will be a Cartesian product of the set S of all the students in this class and the set G of all possible grades
Let S = { Alice, Bob, Chris } and G = { A, B, C }
D = { (Alice, A), (Alice, B), (Alice, C), (Bob, A), (Bob, B), (Bob, C), (Chris, A), (Chris, B), (Chris, C) }
The final grades will be a subset of this: { (Alice, C), (Bob, B), (Chris, A) }
Such a subset of a Cartesian product is called a relation (more on this later in the course)
№35 слайд
Содержание слайда: Set operations: Intersection
Formal definition for the intersection of two sets: A ∩ B = { x | x A and x B }
Further examples
{1, 2, 3} ∩ {3, 4, 5} = {3}
{New York, Washington} ∩ {3, 4} =
No elements in common
{1, 2} ∩ =
Any set intersection with the empty set yields the empty set
№39 слайд
Содержание слайда: Disjoint sets 3
Formal definition for disjoint sets: two sets are disjoint if their intersection is the empty set
Further examples
{1, 2, 3} and {3, 4, 5} are not disjoint
{New York, Washington} and {3, 4} are disjoint
{1, 2} and are disjoint
Their intersection is the empty set
and are disjoint!
Their intersection is the empty set
№41 слайд
Содержание слайда: Set operations: Difference
Formal definition for the difference of two sets:
A - B = { x | x A and x B }
A - B = A ∩ B Important!
Further examples
{1, 2, 3} - {3, 4, 5} = {1, 2}
{New York, Washington} - {3, 4} = {New York, Washington}
{1, 2} - = {1, 2}
The difference of any set S with the empty set will be the set S
№42 слайд
Содержание слайда: Set operations: Symmetric Difference
A symmetric difference of the sets contains all the elements in either set but NOT both
Formal definition for the symmetric difference of two sets:
A B = { x | (x A or x B) and x A ∩ B}
A B = (A U B) – (A ∩ B) Important!
Further examples
{1, 2, 3} {3, 4, 5} = {1, 2, 4, 5}
{New York, Washington} {3, 4} = {New York, Washington, 3, 4}
{1, 2} = {1, 2}
The symmetric difference of any set S with the empty set will be the set S
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