Chapter 2 Exam
⊆
"is a subset of"... ex: x ⊆ A "x is a subset of A" In other words, every element of set X is contained in set A
Number of distinct subsets formula
2^n
Number of proper subsets formula
2^n-1
One to one correspondance
A one to one correspondence between sets (A) and (B) is an assignment, for each element of A, of exactly one element of B in such a way that all elements of B are used.
Null/Empty set
A set with no elements Symbolized by { } or ϕ NOTE: { ϕ } is NOT a null or empty set. This set has a cardinality of 1. The same applies for { 0 }
Natural Numbers
All positive integers (not including zero)
Write in set builder notation: B = {1, 2, 3, 4, 5}
B = {x | x ∈ N and x < 6 }
The three ways a set can be written are blank, blank, and blank
Description, roster form, set builder notation
Proper Subset
If every element of set A is also an element of a set B but A≠B then A is a proper subset of B, or A⊂B
⊂
Is a proper subset
Description of a set example
Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday Description: This set represents the days of the week.
A set that contains no elements is called a(an)
Null Set or Empty Set
A collection of objects is called a(n)
Set
Subset
Set A is a subset of set B, symbolized by A ⊆ B , if and only if all the elements of set A are also elements of set B.
Union
Set containing all elements of two sets, whether common or not to both sets
Equivalent Sets
Sets with the same amount of elements A = { blue, red, yellow, pink } B = { H, 6, O, 3 } Sets (A) and (B) are equivalent
Problem: indicate the set of natural numbers that satisfies the equation "x + 2 = 0"
Since x = -2, and -2 is not a natural number, the answer is ϕ, or { }
Cartesian Product
The Cartesian product of set A and set B, symbolized by A × B and read "A cross B," is the set of all possible ordered pairs of the form ( a , b ) , where a ∈ A and b ∈ B .
The number of elements in a set is called
The cardinal number
Universal Set
The set of all elements under consideration in a given situation, designated by U (italicized)
Set Builder Notation
Type of notation that describes the properties an element must have to be included in a set. Ex: E = { x | x ∈ N and x > 10 } Means "Set E is the set of all the elements x such that x is a natural number and x is greater than 10." So, { 11, 12, 13, 14, ... }
A set which contains all of the elements for a particular discussion is called the
Universal Set
Is a null or empty set a subset of every other set?
Yes
Element / member
a unit in a set ex) 9 is a member of the set of numbers 1-10
Set
collection of objects, which are called elements or members of the set. ex) The USA is a set of 50 states.
Two sets that contain exactly the same elements
equal sets
Two sets that are not identical but which contain the same number of elements is called
equivalent sets
Equal Sets
have EXACTLY the same elements Note: order DOES NOT MATTER, so A = { 1, 2, 3 } B = { 2, 1, 3 } so, A = B
∈
is an element of
∉
is not an element of
Well defined set
its contents can be clearly determined. ex) the set of US presidents is a well defined set, but the set of good US presidents is not
Roster form
lists the elements/members of a set between braces, { }. Things to note: 1) commas separate elements within braces, { } ex) {red, orange, yellow, green, blue, purple} 2) order does not matter within braces ex) {1, 2, 3, 4, 5} is the same as {2, 4, 3, 5, 1} 3) sets are generally named with capital letters, and N (italicized) indicates a set of natural numbers, like {1, 2, 3, 4, 5} 4) Ellipses indicate that elements in a set continue in the same manner, such as {1, 2, 3, 4, 5, ...} indicating 1-infinite
Two sets that share a cardinal number can be placed in a(n)__________ correspondence.
one to one
Cardinal Number
the number of elements in a set n (A) = the number of elements in set A, or, the cardinal number of set A if A = { a, s, d, f, g, h }, then A's cardinality is 6
Complement Set
the set of all elements that belong to the universe U but do not belong a certain set is
The two ways to indicate an empty set are
{ } and ϕ
Answer in roster form: the set of natural numbers between 7 and 12, inclusive.
{7, 8, 9, 10, 11, 12} note: inclusive indicates to include 7 and 12, and not just the natural numbers between 7 and 12.
Answer in roster form: the set of natural numbers between 7 and 12
{8, 9, 10, 11}
Is not a subset of
⊄
