MA110

A Set is defined as a collection of well-defined unique objects, known as elements of the set.
Notation: Sets are denoted by capital letters such as A, B, C, etc.
Elements of a Set are represented by lowercase letters like a, b, c, d, e, f.

If x is an element of set A, we write this as "x belongs to A" or symbolically, x ∈ A.
Conversely, if x is not an element of set A, we express this as "x does not belong to A" or symbolically, x ∉ A.
The symbol "∈" indicates membership, and "∉" indicates non-membership.

There are several methods to express sets:

In the roster method, all elements of the set are listed within brackets and separated by commas.
Example: If B is the set of all days in a week, it is represented as:
- B = {Monday, Tuesday, Wednesday, Thursday, Friday, Saturday, Sunday}.
Another example: The set of all vowels in the English alphabet:
- C = {a, e, i, o, u}.

In the set builder method, we describe the elements of the set by specifying a property that determines the elements uniquely.
Sets using this method are expressed as:
- A = {x | P(x)} or A = {x : P(x)}, which reads as "A is the set of all x such that P(x) is true".
Here, P is a property or condition that elements must satisfy.
Example:
- For Y, the set of all months of a year can be expressed as:
- Y = {x | x is a month of the year}.
- For B, the set of perfect squares of natural numbers:
- B = {x ∈ ℕ | x is a perfect square}.

A Venn diagram is a pictorial representation of a set, often using geometric shapes (like circles, triangles, or rectangles) to represent the sets.
Elements are represented as points enclosed in the geometric figures of the diagram.
Examples in Venn Diagrams:
- Set A: {1, 2, 3}
- Set B: {a, b, c, d, e, f}
- Set C: {4, 5, 6}

If elements are repeated in the listing, they should only be written once.
The order in which elements are listed is immaterial (does not affect membership).

An open interval is defined for two numbers a and b where a < b:
- Defined as: (a, b) = {x | x ∈ ℝ, a < x < b}.
- Numbers between a and b belong to this interval, but a and b themselves do not belong.

A closed interval for a and b where a < b is defined as:
- Defined as: [a, b] = {x | x ∈ ℝ, a ≤ x ≤ b}.
- All numbers between a and b, including a and b, belong to this interval.

A semi-closed interval is either open on one end:
- For [a, b): {x | x ∈ ℝ, a ≤ x < b}.
- For (a, b]: {x | x ∈ ℝ, a < x ≤ b}.

The set of all real numbers greater than a is represented as:
- (a, ∞) = {x | x ∈ ℝ, x > a}.
The set of all real numbers greater than or equal to a is:
- [a, ∞) = {x | x ∈ ℝ, x ≥ a}.

The cardinality of a set A, denoted as n(A), is the number of distinct elements contained in a finite set.
Example: For the set A = {4, 2, 3}, the cardinality is: n(A) = 3.

A set containing no elements is called an empty set or null set, denoted by the symbol Ø or {} or Void Set.
Example: A = {x | x ∈ ℕ, 1 < x < 2} has n(A) = 0.

A set containing only one element is called a singleton set.
Example: Let A be the set containing all integers that are neither positive nor negative:
- A = {0}, thus n(A) = 1.

A set in which the process of counting the elements comes to an end is called a finite set.
Example: If A = {b, e, a, u, t, i, f, l}, then n(A) = 8. Hence, A is a finite set.

Two sets are said to be equal if they contain the same elements:
- Symbolically, A = B if each element of A is in B and vice versa (A ⊆ B and B ⊆ A).
Example: If X is the set of letters in the word "ABBA" and Y is the set of letters in the word "BABA":
- X = {A, B} and Y = {B, A}.
- Thus, X = Y and they are equal sets.

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