Set Theory Concepts.
1.1 Sets:
Definition: A set is a
collection of distinct objects.
Representation:
Roster Method: Listing all elements within
curly braces.
Example: A = {1, 2, 3, 4}
Set-Builder Notation: Defining a set by a property
that its elements must satisfy.
Example: B = {x | x is an even integer}
Types
of Sets
Finite Set: A set with a finite number of
elements.
Example: {a, b, c}
Infinite Set: A set with an infinite number
of elements.
Example: {1, 2, 3, …}
Empty Set (∅ or {}): A set with no elements.
Singleton Set: A set with only one element.
Example: {5}
Universal Set (U): The set of all possible
elements under consideration.
Set
Operations
Union (∪): The union of two sets A and B
is the set of all elements that are in A, or in B, or
in both.
A ∪ B = {x | x ∈ A or x ∈ B}
Intersection (∩): The intersection of two sets A
and B is the set of all elements that are in both A and B.
A ∩ B = {x | x ∈ A and x ∈ B}
Complement: The complement of a set A
(with respect to the universal set U) is the set of all elements in U that are
not in A.
A’ = {x | x ∈ U and x ∉ A}
Difference (A – B): The difference between two
sets A and B is the set of all elements that are in A but not in B.
A – B = {x | x ∈ A and x ∉ B}
Power
Sets
"The power set of a set A is the set
of all possible subsets of A, including the empty set and A
itself. If A has n elements, then the power set of A has 2^n elements."
Example: If A = {a, b}, then the power set of A
is P(A) = { {}, {a}, {b}, {a, b} }
Cartesian
Products
"The Cartesian product of two sets A
and B is the set of all possible ordered pairs (a, b) where a
is in A and b is in B."
A × B = {(a, b) | a ∈ A and b ∈ B}
Example: If A = {1, 2} and B = {x, y}, then A × B
= {(1, x), (1, y), (2, x), (2, y)}