Understanding Sets: Roster & Set-Builder Notation

• A set is a well-defined collection of distinct objects, considered as an object in its own right.
• The objects in a set are called elements or members of the set.
• Sets are typically denoted by uppercase letters (e.g., A, B, C), and elements by lowercase letters (e.g., a, b, c).
• If 'x' is an element of set A, it is written as x ∈ A; If 'x' is not an element of A, it is written as x ∉ A.
• Sets can be defined in several ways: by listing all elements (roster method), or by specifying a property that all elements must satisfy (set-builder notation).

Roster Method

• Listing all elements of a set within curly braces.
• For example, A = {1, 2, 3, 4} defines a set A containing the numbers 1, 2, 3, and 4.
• This method is suitable for finite sets with a manageable number of elements.

Set-Builder Notation

• Defining a set by specifying a property its elements must satisfy.
• Has the form: {x | P(x)}, where P(x) is a predicate (condition) that 'x' must satisfy to be in the set.
• Example: B = {x | x is an even integer} defines the set B of all even integers.

Null Set (Empty Set)

• A set containing no elements.
• Represented by the symbol ∅ or {}.
• It is a subset of every set.

Finite Set

• A set with a finite number of elements.
• It is possible to count all the elements in the set, and the counting process comes to an end.
• Example: A = {1, 2, 3} is a finite set with three elements.

Infinite Set

• A set that is not finite; it contains an infinite number of elements.
• The elements cannot be counted in a finite amount of time.
• Example: The set of natural numbers N = {1, 2, 3, ...} is an infinite set.

Cardinality of Sets

• The cardinality of a set is a measure of the "number of elements" in the set.
• For a finite set, the cardinality is simply the number of elements.
• Denoted by |A| for a set A.
• Example: If A = {1, 2, 3}, then |A| = 3.
• For infinite sets, cardinality is more complex i.e. natural numbers vs real numbers.

Cardinality of Infinite Sets

• Infinite sets can have different cardinalities.
• The "smallest" infinite cardinality is the cardinality of the set of natural numbers, denoted by ℵ₀ (aleph-null).
• Sets with cardinality ℵ₀ are called countably infinite because their elements can be listed in a sequence.
• The set of real numbers has a larger cardinality, denoted by 'c' (for continuum).
• It is uncountably infinite.

Universal Set

• A set that contains all objects under consideration in a given context.
• Denoted by U.
• All other sets are subsets of the universal set.
• The choice of the universal set depends on the specific problem or context.
• For example, when working with numbers, the universal set might be the set of real numbers R.

Subsets

• If every element of a set A is also an element of a set B, then A is a subset of B.
• Written as A ⊆ B.
• Formally: A ⊆ B if for all x, if x ∈ A, then x ∈ B.

Proper Subset

• If A is a subset of B, and A is not equal to B (i.e., B contains at least one element not in A), then A is a proper subset of B.
• Written as A ⊂ B.
• Formally: A ⊂ B if A ⊆ B and A ≠ B.

Equality of Sets

• Two sets A and B are equal if they contain exactly the same elements.
• Written as A = B.
• Formally: A = B if and only if A ⊆ B and B ⊆ A.

Power Set

• The set of all possible subsets of a set A, including the empty set and A itself.
• Denoted by P(A) or 2ᴬ.
• If A is a finite set with n elements, then the power set P(A) has 2ⁿ elements.
• Example: If A = {a, b}, then P(A) = {∅, {a}, {b}, {a, b}}.

Disjoint Sets

• Two sets A and B are disjoint if they have no elements in common.
• Formally: A and B are disjoint if A ∩ B = ∅ (where ∩ denotes intersection).

Overlapping Sets

• Two sets A and B are overlapping if they have at least one element in common
• Formally: A and B are overlapping if A ∩ B ≠ ∅ (where ∩ denotes intersection).