set_theory_flashcards
Set {}
An unordered collection of distinct objects, e.g., {apple, orange, banana}.
Element of a Set ∈
An item in a set, denoted as ‘a ∈ A’ meaning ‘a is in A’.
Empty Set Ø or {}
A set with no elements, denoted as Ø or {}.
Finite Set
A set with a limited number of elements, e.g., {1, 2, 3}.
Infinite Set
A set with an unlimited number of elements, e.g., {1, 2, 3, …}.
Subset ⊆
A set where all elements are also in another set, denoted as A ⊆ B, e.g., {1, 2} ⊆ {1, 2, 3}.
Proper Subset ⊂
A subset that is not equal to the set it is contained within, e.g., {1, 2} ⊂ {1, 2, 3}.
Superset ⊇
A set containing all elements of another set, e.g., {1, 2, 3, 4} ⊇ {1, 2}.
Universal Set
The set that contains all possible elements in a given context.
U={1,2,3,4}
Complement
U={1,2,3,4}, A={1,2}
What is A’?
All elements not in a specified set, relative to the universal set, e.g., if U={1,2,3,4}, A={1,2}, then A’={3,4}.
Union
{1,2,3} ∪ {2,3,4,5}
All elements from two sets, without ones that appear in both denoted A ∪ B, e.g., {1,2} ∪ {2,3} = {1,2,3,4,5}.
Intersection
{1,2,3} ∩ {2,3,4}
Elements common to both sets, denoted A ∩ B, e.g., {1,2,3} ∩ {2,3,4} = {2,3}.
Difference
{1,2,3} - {2,3}
Elements in one set but not the other, denoted A - B, e.g., {1,2,3} - {2,3} = {1}.
Symmetric Difference
{1,2} ⊕ {2,3}
Elements in either set but not both, denoted A ⊕ B, e.g., {1,2} ⊕ {2,3} = {1,3}.
Commutative Law (Union)
A ∪ B =?
A ∪ B = B ∪ A.
Commutative Law (Intersection)
A ∩ B = ?
A ∩ B = B ∩ A.
Associative Law (Union)
(A ∪ B) ∪ C = ?
(A ∪ B) ∪ C = A ∪ (B ∪ C).
Associative Law (Intersection)
(A ∩ B) ∩ C =?
(A ∩ B) ∩ C = A ∩ (B ∩ C).
Distributive Law (Intersection over Union)
A ∩ (B ∪ C) =?
A ∩ (B ∪ C) = (A ∩ B) ∪ (A ∩ C).
Distributive Law (Union over Intersection)
A ∪ (B ∩ C) =?
A ∪ (B ∩ C) = (A ∪ B) ∩ (A ∪ C).
Identity Law (Union)
A ∪ Ø = ?
A ∪ Ø = A.
Identity Law (Intersection)
A ∩ U =?
A ∩ U = A.
Complement Law (Union)
A ∪ A’ = ?
A ∪ A’ = U.
Complement Law (Intersection)
A ∩ A’ = ?
A ∩ A’ = Ø.
Idempotent Law (Union)
A ∪ A =?
A ∪ A = A.
Idempotent Law (Intersection)
A ∩ A =?
A ∩ A = A.
Double Complement Law
(A’)’ =?
(A’)’ = A.
De Morgan’s Law (Complement of Union)
(A ∪ B)’ = ?
(A ∪ B)’ = A’ ∩ B’.
De Morgan’s Law (Complement of Intersection)
(A ∩ B)’ =?
(A ∩ B)’ = A’ ∪ B’.
Cartesian Product
if A={1,2} and B={x,y}, then what is A × B?
The set of all ordered pairs from two sets, e.g., if A={1,2} and B={x,y}, then A × B = {(1,x), (1,y), (2,x), (2,y)}.
Power Set
if A = {1, 2}, then 𝒫(A) is?
The set of all subsets of a set, including the empty set, e.g., if A = {1, 2}, then 𝒫(A) = {Ø, {1}, {2}, {1, 2}}.
