Set Theory Flashcards
Define a set
A set is an unordered collection of objects
What is the rooster method
S = {a, b, c, d} denotes the set S containing elements a, b, c, and d.
- Ellipses can be used when pattern is clear: S = {a, b, c, d, . . . , z}
Explain this set-builder notation: S = {x E z+| x is odd and x < 10)
For some positive integers, x is odd and x < 10. Contains all the elements from this set that satisfy the predicate
What are the 7 important sets
- N = {0, 1, 2, 3, . . . }: set of natural numbers, numbers > 0
- Z = {. . . , −2, −1, 0, 1, 2, . . . }: set of integers
- Z + = {1, 2, 3, . . . } set of positive integers
- Q = { p q | p ∈ Z, q ∈ Z, and q ̸= 0} set of rational numbers
- R: set of real numbers
- R +: set of positive real numbers
- C: set of complex numbers
Define a universal set (U)
The universal set is the set containing everything under consideration
Define an empty set (∅)
an empty set is a set with no elements
How is cardinality shown for A
|A|
When is there set equality
when sets have the same elements,
A = B ↔ ∀x. (x ∈ A ↔ x ∈ B)
What does this mean: (A ⊆ B)
set A is a subset of B, if and only if every element of A is also an element of B: ∀x. (x ∈ A → x ∈ B)
If A ⊆ B, but A ̸= B what do we call this subset
this is called a proper subset denoted by A ⊂ B
Provide an example of a subset
- The set of all computer science students is a subset of all students.
- The set of integers with squares less than 100 is not a subset of N.
What is the given rule to calculate how many sub-sets any given set has
2 to the power of n (2^n), where n is the cardinality of the set n = |A|
What is the cardinality of the set: A = {1 , 2, 3}
2^3 = 8
What does this symbol denote (∈)
“is in”, it denotes a set membership
What is the unions of set A and B denoted by
A and B, denoted by A ∪ B, is the set:
{x | x ∈ A ∨ x ∈ B}
Complete this example of what the union symbol does {1, 2, 3} ∪ {3, 4, 5}
{1, 2, 3} ∪ {3, 4, 5} = {1, 2, 3, 4, 5}
What is the intersection of set A and B denoted by
The intersection of the sets A and B, denoted by A ∩ B, is the set:
{x | x ∈ A ∧ x ∈ B}
Complete these examples of what the intersection symbol does:
- {1, 2, 3} ∩ {3, 4, 5}
* {1, 2, 3} ∩ {4, 5, 6}
- = {3}
- = ∅
What is the difference of set A and B denoted by
The difference of the sets A and B, denoted by A − B, is the set:
{x | x ∈ A ∧ x ∈/ B}
Complete these examples of what the difference symbol does:
- {1, 2, 3} − {3, 4, 5}
* {1, 2, 3} − {4, 5, 6}
- {1,2}
- {1,2,3}
What is the complement of set A and B denoted by
The complement of the set A (with respect to U), denoted by A, is the set:
U − A = {x | x ∈ U ∧ x ∈/ A}
Complete these examples of what the complement symbol does
What does the | symbol mean
“such that for all elements of x”
What does the line above the symbols mean
The complement of
What does this mean A ⊆ B
A is a subset of B
How do I know the cardinality
The number of numbers in a set
Define a power set
The set of all subsets of a particular set
e.g. A = {1 , 2, 3}
P(A) = {∅, {1}, {2}, {3}, {1,2}, {1,3}, {2,3}, {1,2,3}
What does this mean [a, b]
{x | a ≤ x ≤ b}
What does this mean [a, b)
{x | a ≤ x < b}
What does this mean (a, b]
{x | a < x ≤ b}
What does this mean (a, b)
{x | a < x < b}
How is a union of sets A and B denoted
A U B
How is an intersection of sets A and B denoted
A n B
How is the difference of sets A and B denoted
A - B
How is the complement of sets A and B denoted
A’ (there should be a line above)
State the Identity Law
A U ∅ = A, A n U = A
State the Domination law
A U U = U, A n ∅ = ∅
State the Complementation law
A’’ = A
State the Commutative law
A U B = B U A , A n B = B n A (Change the ordering)
State the Associative law
A U (B U C) = (A U B) U C, same for n (change the bracketing)
State the Distributive law
A n (B U C) = (A n B) U (A n C ), same for A U ( B n C)
State De Morgan’s law
(A U B)’ = A’ n B’, (A n B)’ = A’ U B’
State the Absorption law
A U ( A n B) = A, A n ( A U B) = A
State the Complement law
A. U A’ = U, A n A’ = ∅
