Sets
What is the formula for for finding out the number of elements/cardinality in P(A) or Power Set?
$2^n$
What is the name of the set with no elements?
**Empty set** or **null set**.
⊄ or ⊆ **{4} ⊆ {2, 4, 6, 10}**
⊆
⊄ or ⊆ {3, 12, 5, 19} and {19, 3, 5, 12}
⊆
How are power sets denoted?
P(A)
If set A = {-9, 13, 6}, what is the P=(A) or Power Set?
P(A)={ϕ, {-9}, {13}, {6}, {-9,13}, {13,6}, {6,-9}, {-9,13,6}}
This type of set is the set that contains all of the subset of the set.
Power set
Why are empty sets a subset of every set?
Because all sets have empty sets.
What kind of letters are used to name a set?
Capital letters
It is the number of elements in a set.
Cardinal number
Set I is the set of months with 35 days.
I = { } or I = ∅
Set M is the set of cars with 60 doors.
I = { } or I = ∅
How can you determine if whether or not set A is a subset of set B?
If all elements on set A is in set B, then set A is a subset of set B.
What does A/B or A-B denote?
It denotes **relative complement**.
What does $A^{c}$ denote?
It denotes complement.
What does ∆ denote in Set Theory?
It denotes symmetric difference.
What is the name of this ⊂ symbol?
It is called a proper subset/strict subset.
How is ∈ read in Set Theory and when do you use it?
It is read as "element of" and you use it if an element is a member of the set.
What does complement in Set Theory mean?
It means all the elements that do not belong to the set. Example: $A^c$ means you're looking for the complement of A. It basically means you need to know all of the elements except for the ones that are in A.
Sets must be well-defined. What do they mean by this?
It means that it should be clear what the elements in that set are. We should easily know what is counted as an element, and what isn't.
These are the objects within the set.
Elements.
It is contrary to the finite set.
Infinite Set
What is the name of this ⊄ symbol?
Not a subset.
Set T is the set of counting numbers between 1 and 2. What are its elements?
**T = { }** or **T =** **Ø**
What is the symbol for an empty set, or a null set?
**{ }** or **Ø**
What is used to separate elements within set?
, or commas
If a set contains many elements, we often use three dots. What do we call those three dots?
... or ellipsis
Given any two sets A and B, if every element in A is also an element in B, what is the relationship between A and B?
A is a subset of B.
How is **A ⊆ B** read?
A is a subset of B.
If a set doesn't have any elements, it is known as...
Empty set or null set
What does = denote in set theory?
Equality
What is the difference between Equivalent sets and Equal sets?
Equivalent sets are equal in **number of elements**, while equal sets are **equal in elements**.
If a set contains no element or a definite number of elements, what kind of set is it?
Finite Set
What kind of set is A = {x : x is a month in a year}?
Finite set, because it is clear that the set has a finite number of element—12.
What are the possible subsets of **S = {3, 6, 9, 12}**?
Four Elements: {3, 6, 9, 12} Three Elements: {3, 6, 9}, {3, 6, 12}, {3, 9, 12}, {6, 9, 12} Two Elements: {3, 6}, {3, 9}, {3, 12}, {9, 12} One Element: {3}, {6}, {9}, {12} Zero Elements: { } or Ø
Who initiated the concept of set theory?
Georg Cantor
Does the order of elements matter?
No
"The set of famous dancers." Is this a well-defined set?
No, because people will have different point of views on which dancers are famous—which makes it difficult to know which elements truly are part of the set, and which elements aren't.
Does the number of times an element reoccurs matter?
No, if an element occurs more than once it is counted only once.
If a finite set is non-empty, what do you call it?
Non-empty finite set.
What does relative complement mean?
Relative complement is usually denoted as A/B. If put in this way, A/B or A-B is the elements of A but without the elements of B. 
In this form of Representation of Sets, all the elements of the set are listed, separated by commas and enclosed between curly braces { }.
Roster Form
In what form is A ={1996,2000,2004,2008,2012} listed?
Roster Form
What are the two ways that you can represent sets?
Roster Form (or Tabular form) and Set Builder Form
It is a well-defined group or collection of objects that share common characteristics.w
Set
In this form of set, all elements have a common property. This property is not applicable to the objects that do not belong to the set.
Set Builder Notation
What form of set is S={ x: x is an even prime number}?
Set Builder Notation
What kind of set is B = {y : y is a whole number which is not a natural number}?
Singleton
If a set contains only one element, what do you call it?
Singleton Set
What kind of set is A = {x : x is an even prime number}?
Singleton Set
Find the cardinality of the following set. Set D is the set of vowels in English alphabet.
Solution: D = {𝑎, 𝑒, 𝑖, 𝑜, 𝑢} Answer: n(D) = 5
Find the cardinality of the following set. Set E is the set of letters in the word "survivor".
Solution: E = {𝑠, 𝑢, 𝑟, 𝑣, 𝑖, 𝑜 } Answer: n(E) = 6
Find the cardinality of the following set. Set K is the set of counting numbers less than 5.
Solution: K = {1, 2, 3, 4} Answer: n(K) = 4
Find the cardinality of the following set. Set M is the set of odd numbers between 1 and 3.
Solution: M = { } or M = ∅ Answer: n(M) = 0
Find the cardinality of the following set. Set R is the set of letters in the word "difficulty".
Solution: R = {𝑑, 𝑖, 𝑓, 𝑐, 𝑢, 𝑙,𝑡, 𝑦} Answer: n(R) = 8
What does this symbol **⊆** denote?
Subset
What kind of set is A = {x : x is a natural number}?
There are infinite natural numbers. Hence, A is an **infinite set.**
What kid of set is B = {y: y is the ordinate of a point on a given line}?
There are infinite points on a line. So, B is an infinite set.
Describe Equal Sets.
They are **equal in elements**.
Describe Equivalent sets.
They are two sets are equivalent if they have **same number of elements.
What are the possible subsets of **R = {1, 2}**?
Two Elements: **{1, 2}** or **{2, 1}**. One Element: **{1}**, **{2}**. Zero Element: **{ }** or **Ø**
It is the set that contains all objects under consideration.
Universal Set or U
What does U denote?
Universal set
"The set of all even numbers." Is this a well-defined set?
Yes, because we can easily deduce that the elements in the set are E = { 2, 4, 6, ... }.
Can set A be a subset of itself?
Yes. A subset is a set where all of its elements are in the other set. Since set A has all of the element of set A, it is a subset of itself.
When do you use = or Equality in Set Theory?
You use it if both sets have the same elements. Example: A = {1, 2, 3} B = {1, 2, 3} A = B
When do you use ⊂ or proper subset/strict subset?
You use it if the other set has more elements than the subset. Example: ****A = **{7, 13, 15, }** B = **{1, 7, 9, 13, 15, 23} A ⊂ B = {7, 13, 15 }** ⊂ **{1, 7, 9, 13, 15, 23}** Set A is a proper subset/strict subset of B because it has less elements than B. Strict set = there are things in the bigger set that aren't in the smaller set.
When do you use ∆ or symmetric difference mean in Set Theory?
You use it to find that belong to A or B but **not to their intersection.** Example: If A = {1, 2, 3, 4} and B = {2, 4, 6, 8} A ∆ B = {1, 3, 6, 8}
When do you use this ⊄ symbol?
You use it when the left set is not a subset of right set.
What do we use to group the elements of the set?
curly brackets or { }
How is the cardinal number of set A denoted?
n(A)
State whether each of the following sets is well-defined or not. "The set of tasty food."
not
State whether each of the following sets is well-defined or not. "The set of versatile actress."
not
State whether each of the following sets is well-defined or not. "The set of young politicians."
not
Equivalent sets require you to have equal...
number of elements
State whether each of the following sets is well-defined or not. "The set of planets in our solar system."
well-defined
State whether each of the following sets is well-defined or not. "The set of types of matter."
well-defined
⊄ or ⊆ **{𝑏, ℎ, 𝑟, 𝑞} and {ℎ, 𝑟}**
⊄
**∅ and {1, 3, 5, 7}**
⊆
⊄ or ⊆ **{3, 5, 7} and {1, 2, 3, ... }**
⊆
