What are Sets? | Set Theory

March 27th, 2024

Sets

A **set** is a well-defined collection of distinct objects, considered as an object in its own right. These objects can be anything: numbers, letters, symbols, or even other sets. T2=The objects in a set are called its elements or members. Sets are typically denoted by curly braces {}, and the elements are listed inside separated by commas.

For example:

- {1, 2, 3, 4, 5} is a set containing the numbers 1, 2, 3, 4, and 5.

- {a, b, c, d} is a set containing the letters a, b, c, and d.

- {apple, orange, banana} is a set containing the names of fruits.

Sets are defined by their elements and are not ordered. This means that the order in which elements are listed in a set does not matter, and each element appears only once in T2=the set (sets do not contain duplicate elements).

Sets are fundamental in various branches of mathematics, including algebra, calculus, and discrete mathematics, and they serve as a basis for defining other mathematical concepts like functions, relations, and operations.

Empty Sets

An **empty**, set also known as the **null** set, is a set that contains no elements. In other words, it is a set with a cardinality of zero. The notation used to denote the empty set is usually \(\emptyset\) or {} (curly braces with nothing inside).

For example:

- \(\emptyset\) represents the empty set.

- {} also represents the empty set.

It's important to note that the empty set is still a valid set, even though it contains no elements. It is a fundamental concept in set theory and mathematics in general because it helps define certain properties of sets and is often used in mathematical proofs and definitions.

Finite Sets

A **finite** set is a set that contains a specific number of elements, where this number can be counted or determined. In other words, a set is finite if it has a definite endpoint and can be enumerated. For example:

- {1, 2, 3, 4, 5} is a finite set containing 5 elements.

- {apple, orange, banana} is a finite set containing 3 elements.

Infinite Sets

An **infinite** set is a set that contains an unlimited (or infinitely many) number of elements. These sets do not have a definite endpoint, and their elements cannot be counted exhaustively. For example:

- The set of all natural numbers N = {1,2,3,4,...} is an infinite set because it continues indefinitely.

- The set of all integers Z = {...,−3,−2,−1,0,1,2,3,...} is also infinite.

Equal Sets

For example:

- If set A = {1, 2, 3} and set B = {1, 2, 3}, then A and B are equal sets because they contain the same elements.

- Similarly, if set C = {3, 2, 1}, then C is also equal to A and B because it contains the same elements, just listed in a different order.

Mathematically, we denote equal sets using the symbol "=". So, if A and B are equal sets, we write A = B.

It's important to note that for sets to be equal, they must have exactly the same elements. If even one element is different between two sets, they are not equal sets.

Subsets

In set theory, a **subset** is a set that contains only elements that are also elements of another set. In other words, if every element of set A is also an element of set B, then A is a subset of B.

Formally, if A and B are sets, we say that A is a subset of B, denoted as \(A \subseteq B\), if every element of A is also an element of B. If A is a subset of B but A is not equal to B, then A is called a proper subset of B, denoted as \(A \subset B\).

For example:

- Let A = {1, 2} and B = {1, 2, 3, 4}. Here, A is a subset of B because every element of A (1 and 2) is also an element of B.

- Let C = {3, 4, 5} and D = {1, 2, 3, 4, 5}. In this case, C is a proper subset of D because every element of C is also an element of D, but C is not equal to D.

Some important properties of subsets include:

- Every set is a subset of itself: A ⊆ A.

- The empty set ∅ is a subset of every set: ∅ ⊂ A for any set A.

- If A = B, then A is a subset of B and B is a subset of A: A = B ⇒ A ⊆ B and B ⊆ A.

Some common subsets of the set of real numbers, particularly intervals, along with their notations:

Open Interval:

An open interval between two real numbers a and b (where a < b) consists of all real numbers between a and b excluding a and b themselves.

- Notation: (a,b)

- Example: The open interval between 0 and 1 is (0,1), which includes all real numbers between 0 and 1, but neither 0 nor 1.

Closed Interval:

A closed interval between two real numbers aa and bb consists of all real numbers between aa and bb including aa and bb themselves.

- Notation: [a,b]

- Example: The closed interval between 0 and 1 is [0,1], which includes all real numbers between 0 and 1, including 0 and 1.

Half-Open or Half-Closed Intervals:

These intervals include one endpoint but not the other.

- a. Half-open interval: [a,b) or (a,b]

- Example: [0,1) includes all real numbers greater than or equal to 0 and less than 1.

Unbounded Intervals:

These intervals extend indefinitely in one or both directions.

a. Interval with one endpoint at infinity: (a,∞) or (−∞,a)

- Example: (0,∞) includes all real numbers greater than 0.

b. Interval with both endpoints at infinity: (−∞,∞)

- Example: (−∞,∞) includes all real numbers.

Singleton Set:

A set containing only one real number.

- Notation: {a}

- Example: {0} is a singleton set containing only the real number 0.

Power Set:

The power set of a set S is the set of all subsets of S, including the empty set and S itself. In other words, it is the set of all possible combinations of elements from S.

- Notation: If S is a set, then its power set is denoted by P(S) or P(S).

- Example: If S={a,b} S={a,b}, then the power set of S is (P(S) = {∅, {a}, {b}, {a, b}}).

Universal Set:

The universal set, denoted by U or Ω, is the set that contains all the elements under consideration in a particular discussion or context. It serves as the "universe" from which all other sets are derived or defined.

- Example: In a classroom setting where we are studying geometry, the universal set might be the set of all geometric shapes that can be studied in that class, such as triangles, circles, squares, etc.

Venn diagram

A **Venn diagram** is a graphical representation used to illustrate the relationships between different sets. It consists of circles (or other shapes) that represent sets, with overlapping regions indicating the intersection of sets. Venn diagrams are named after the mathematician **John Venn**, who introduced them in the late 19th century.

In a **Venn diagram**, each circle typically represents a different set. The elements of each set are represented by points within the respective circle. The overlapping regions of the circles represent elements that belong to more than one set, showing the intersection of those sets.

Venn diagrams can illustrate various set operations and relationships, including unions, intersections, complements, and more. They are widely used in mathematics, logic, statistics, computer science, and other fields to visualize and analyze relationships between different sets or groups.

For example, consider a Venn diagram representing the sets A and B:

- The circle labeled A represents the elements of set A.

- The circle labeled B represents the elements of set B.

- The overlapping region represents the elements that belong to both sets A and B, showing their intersection.

- The non-overlapping parts of each circle represent elements that belong exclusively to one set but not the other.

Venn diagrams are valuable tools for understanding and **solving problems** involving set theory, logic, and categorical relationships. They provide a visual representation that helps make complex relationships more understandable and accessible.

Union and Intersection of a set

In set theory, union and intersection are two fundamental operations that can be performed on sets.

Union (∪):

- The union of two sets A and B, denoted as A ∪ B, is the set of all elements that are in A, in B, or in both A and B.

- In other words, if you take all the elements from both sets and remove any duplicates, you have the union.

- Symbolically, if A={1,2,3} and B={3,4,5}, then A∪B={1,2,3,4,5}.

Intersection (∩):

- The intersection of two sets A and B, denoted as A ∩ B, is the set of all elements that are common to both A and B.

- In other words, it's the set containing all the elements that belong to both sets A and B.

- Symbolically, if A={1,2,3} and B={3,4,5}, then A∩B={3}.

To summarize:

- Union includes all elements from both sets, eliminating duplicates.

- Intersection includes only elements that are common to both sets.

Difference of two sets

The difference of two sets A and B, denoted as A - B or A \ B, is the set of elements that are in set A but not in set B. In other words, it includes all elements that belong to set A but do not belong to set B.

Symbolically, if A={1,2,3} and B={3,4,5}, then A−BA−B or A∖B is {1,2}, because 1 and 2 are in set A but not in set B. Similarly, B−Aor B∖A would be {4,5}, because those elements are in set B but not in set A.

In set theory, the difference operation is sometimes referred to as the relative complement operation.

Complement of a set

In set theory, the complement of a set A, denoted as A', is the set of all elements that are not in set A but are in the universal set, which is typically the set of all possible elements in the context.

A′={x ∈ U } A′= {x ∉ A}

In simpler terms, the complement of set A contains all elements that are in the universal set but not in set A.

- For example, if the universal set is the set of integers from 1 to 10 (U={1,2,3,4,5,6,7,8,9,10}), and

- set A is the set of even numbers from 1 to 10 (A={2,4,6,8,10}),

- then the complement of set A, denoted as A′A′, would be the set of odd numbers from 1 to 10 (A′={1,3,5,7,9}).

Properties of Complement sets

The complement of a set has several properties that are important to understand:

Complement of the Complement: The complement of the complement of a set A is the set A itself.

- Symbolically, (A′)′ = A.

Complement of the Universal Set: The complement of the universal set is the empty set.

- Symbolically, if U is the universal set, then U′ = ∅.

Complement of the Empty Set: The complement of the empty set is the universal set.

- Symbolically, if ∅ is the empty set, then ∅′ = U.

De Morgan's Laws:

- First Law: The complement of the union of two sets is equal to the intersection of their complements. Symbolically, (A∪B)′ = A′∩B′.

- Second Law: The complement of the intersection of two sets is equal to the union of their complements. Symbolically, (A∩B)′ = A′∪B′.

Complement and Set Operations:

- Complement distributes over set union: A∪A′ = U.

- Complement distributes over set intersection: A∩A′ = ∅.

- Complement of the empty set: ∅′ = U.

- Complement of the universal set: U′ = ∅.

Understanding these properties can help in simplifying expressions involving complements and sets, and they play a crucial role in various areas of mathematics, especially in set theory and logic.

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