BASIC SET THEORY

In the realm of mathematics, set theory serves as a key building block. This branch of logic delves into collections of objects, not limited to just numbers or shapes. While any kind of item can be grouped into a set, mathematicians primarily concentrate on sets relevant to their field. In fact, the language of set theory forms the foundation for defining most mathematical concepts. Crucial elements within this framework include sets themselves, their members, subsets, unions, intersections, and complements. Pioneered by German mathematician Georg Cantor, set theory also lays the groundwork for understanding relations and functions within mathematics

What is Basic Set Theory?

Basic Set Theory is a branch of mathematical logic that studies sets, which are collections of distinct objects. It provides a foundation for various branches of mathematics and is widely used in different areas, including algebra, calculus, and probability theory. Here are some fundamental concepts in basic set theory:

Set:
- A set is a well-defined collection of distinct objects, considered as an object in its own right. For example, the set of natural numbers less than 10 is ${1, 2, 3, 4, 5, 6, 7, 8, 9}$ .
Element:
- An element is an individual object that belongs to a set. In the set mentioned above, 3 is an element of the set.
Representation:
- Sets can be represented using curly braces. For example, the set of even numbers less than 10 can be represented as ${2, 4, 6, 8}$ .
Equal Sets:
- Two sets are equal if they have exactly the same elements, regardless of the order. For example, ${1, 2, 3}$ and ${3, 1, 2}$ are equal sets.
Subset:
- A set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$ . The notation $A \subseteq B$ denotes that $A$ is a subset of $B$ .
Union:
- The union of two sets $A$ and $B$ , denoted by $A \cup B$ , is the set of elements that are in $A$ , or $B$ , or in both.
Intersection:
- The intersection of two sets $A$ and $B$ , denoted by $A \cap B$ , is the set of elements that are common to both $A$ and $B$ .
Complement:
- The complement of a set $A$ , denoted by $A^{'}$ or $A^{c}$ , is the set of all elements in the universal set that are not in $A$ .
Universal Set:
- The universal set, denoted by $U$ , is the set that contains all the elements under consideration

Symbols in Basic Set Theory

In basic set theory, various symbols are used to represent operations and relationships between sets. Here are some common symbols used in basic set theory:

or ${}$ :
- The symbol $\emptyset$ represents the empty set, which is a set with no elements.
$\in$ :
- The symbol $\in$ is used to denote that an element belongs to a set. For example, $a \in A$ means that element $a$ is in set $A$ .
:
- The symbol $\in /$ is used to denote that an element does not belong to a set. For example, $b \in / A$ means that element $b$ is not in set $A$ .
or :
- The symbol $\subset$ or $\subseteq$ denotes that one set is a subset of another. $A \subset B$ means that every element of $A$ is also an element of $B$ , and $A \subseteq B$ allows for the possibility that $A$ is equal to $B$ .
or :
- The symbol $\supset$ or $\supseteq$ denotes that one set is a superset of another. $A \supset B$ means that every element of $B$ is also an element of $A$ , and $A \supseteq B$ allows for the possibility that $A$ is equal to $B$ .
:
- The symbol $\cup$ denotes the union of two sets. $A \cup B$ is the set of elements that are in $A$ , or $B$ , or in both.
:
- The symbol $\cap$ denotes the intersection of two sets. $A \cap B$ is the set of elements that are common to both $A$ and $B$ .
or $\$ :
- The symbols $∖$ or $\$ represent the set difference. $A ∖ B$ or $A \ B$ is the set of elements in $A$ but not in $B$ .
$∁$ or ^{$^{c}$}:
- The symbols $∁$ or $^{c}$ denote the complement of a set. $A^{c}$ is the set of all elements in the universal set that are not in $A$ .
$\times$ :
- The symbol $\times$ is sometimes used to denote the Cartesian product of two sets. $A \times B$ is the set of all ordered pairs $(a, b)$ where $a$ is in $A$ and $b$ is in $B$ .

Basic Set Theory Formulas

In basic set theory, there are several formulas and properties that describe relationships and operations between sets. Here are some fundamental formulas:

Cardinality of a Set:
- The cardinality of a set $A$ , denoted as $∣ A ∣$ , is the count of elements in $A$ . $∣ A \cup B ∣ = ∣ A ∣ + ∣ B ∣ - ∣ A \cap B ∣$
- n(A ∪ B) = n(A) + n(B) - n(A ∩ B) - This formula calculates the cardinality of the union (A ∪ B) of two sets A and B, where n(A) and n(B) are the respective cardinalities of A and B and n(A ∩ B) is the cardinality of their intersection.
- n(A ∩ B) = n(A) + n(B) - n(A ∪ B) - This formula finds the cardinality of the intersection (A ∩ B) of two sets, based on their individual cardinalities and the cardinality of their union.
- n(A - B) = n(A) - n(A ∩ B) - This formula determines the cardinality of the difference (A - B) of set A and set B, where n(A) is the cardinality of A and n(A ∩ B) is the cardinality of their intersection.
- n(∅) = 0 - This formula simply states that the empty set (∅) has no elements, its cardinality is 0

Complement of a Set:
- The complement of a set $A$ with respect to the universal set $U$ , denoted $A^{c}$ , consists of all elements in $U$ that are not in $A$ .

A \cup A^{c} = U

Set Difference:
- The set difference between sets $A$ and $B$ , denoted $A ∖ B$ or $A - B$ , is the set of elements in $A$ but not in $B$ . $A ∖ B = A \cap B^{c}$
De Morgan's Laws:
- De Morgan's laws express the complement of the union and intersection of sets. $(A \cup B)^{c} = A^{c} \cap B^{c}$
- (A∩B)c=Ac∪Bc
Cartesian Product:
- The Cartesian product of two sets $A$ and $B$ , denoted $A \times B$ , is the set of all ordered pairs $(a, b)$ where $a$ is in $A$ and $b$ is in $B$ . $∣ A \times B ∣ = ∣ A ∣ \times ∣ B ∣$
Power Set:
- The power set of a set $A$ , denoted as $P (A)$ , is the set of all subsets of $A$ , including the empty set and $A$ itself. $∣ P (A) ∣ = 2^{∣ A ∣}$
Set Identities:
- Identity laws express the relationship of a set with itself under union and intersection operations.
- $A \cup \emptyset = A$ , $A \cap U = A$

What is a Set?

A set is a collection of distinct elements, considered as an object in its own right. For example, the numbers 1, 2, and 3 are distinct elements when considered separately, but when they are considered collectively as the set {1, 2, 3}, they form a single entity. Sets are typically denoted by curly braces, and the elements are listed inside the braces.

Here's an example:

$A = {1, 2, 3, 4, 5}$

This is a set named $A$ containing the elements 1, 2, 3, 4, and 5. The order in which elements are listed in a set doesn't matter, and each element is unique within the set.

Sets are fundamental in various branches of mathematics, including set theory, which studies the properties and relationships of sets. Sets can be manipulated using operations such as union, intersection, and complement. Additionally, sets play a crucial role in various mathematical structures and concepts, including functions, relations, and logic

Types of Sets

Sets can be classified into various types based on their properties and elements. Here are some common types of sets:

Finite Set:
- A set that contains a definite number of elements.
- Example: $A = {1, 2, 3, 4, 5}$ .
Infinite Set:
- A set that contains an infinite number of elements.
- Example: $B = {1, 2, 3, \dots}$ (the set of natural numbers).
Empty Set (Null Set):
- A set with no elements.
- Denoted by ${}$ or $\emptyset$ .
Singleton Set:
- A set that contains only one element.
- Example: $C = {7}$ .
Equal Set:
- Two sets are equal if and only if they have exactly the same elements.
- Example: $D = {1, 2, 3}$ and $E = {3, 2, 1}$ are equal sets.
Subset:
- Set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$ .
- Denoted by $A \subseteq B$ .
- Example: If $A = {1, 2}$ and $B = {1, 2, 3}$ , then $A$ is a subset of $B$ .
Power Set:
- The set of all subsets of a given set.
- If $S = {a, b}$ , then the power set of $S$ is ${{}, {a}, {b}, {a, b}}$ .
Universal Set:
- The set that contains all the elements under consideration in a particular discussion or problem.
- Often denoted by $U$ .
Complement Set:
- The complement of set $A$ with respect to the universal set $U$ is the set of all elements in $U$ that are not in $A$ .
- Denoted by $A^{'}$ or $A^{c}$ .
Disjoint Set:
- Two sets are disjoint if they have no common elements.
- Example: If $P = {1, 2}$ and $Q = {3, 4}$ , then $P$ and $Q$ are disjoint sets

Methods of Writing Sets

Sets can be represented in various ways, and different methods are used to write and describe sets. Here are some common methods of writing sets:

Roster or Enumeration Method:
- The elements of the set are listed explicitly, separated by commas, and enclosed in curly braces.
- Example: $A = {1, 2, 3, 4, 5}$ .
Set-Builder or Rule Method:
- The set is defined by specifying a rule or condition that its elements must satisfy.
- Example: $B = {x ∣ x is an even number}$ .
Interval Notation:
- Used when dealing with sets of real numbers.
- Example: $C = {x ∣ 1 \leq x \leq 5}$ can be written as $[1, 5]$ .
Venn Diagrams:
- Graphical representation of sets using overlapping circles.
- Common elements are placed in the overlapping region.
- Example: Consider sets $A = {1, 2, 3}$ and $B = {3, 4, 5}$ , their Venn diagram would show the overlapping element $3$ .
Interval Notation for Inequalities:
- Used for sets of real numbers expressed as intervals.
- Example: $D = {x ∣ x > 2}$ can be written as $(2, \infty)$ .
Algebraic Notation:
- Used to express sets in terms of variables and algebraic operations.
- Example: $E = {x ∣ x^{2} - 1 = 0}$ represents the set ${- 1, 1}$ .
Descriptive Method:
- Sets can be described in natural language.
- Example: $F$ is the set of prime numbers less than 10.
Power Set Notation:
- Denotes the power set of a set $A$ , which is the set of all subsets of $A$ .
- Example: If $A = {1, 2}$ , the power set of $A$ is ${1,2}}$ .

Solved examples of Basic Set Theory

Example 1:

Consider two sets: $A = {1, 2, 3, 4, 5}$ $B = {3, 4, 5, 6, 7}$

a. Union (A ∪ B): $A \cup B = {1, 2, 3, 4, 5, 6, 7}$

b. Intersection (A ∩ B): $A \cap B = {3, 4, 5}$

c. Difference (A - B): $A - B = {1, 2}$

d. Complement of A (with respect to the universal set U):

$A^{'} = U - A$

$U = {1, 2, 3, 4, 5, 6, 7, 8, 9, 10}$

$A^{'} = {6, 7, 8, 9, 10}$

Example 2:

Consider sets: $P = {2, 4, 6, 8, 10}$ $Q = {3, 6, 9, 12}$

a. Subset (P is a subset of Q?):

$P \subseteq Q is false, as not every element of P is in Q$

b. Disjoint sets (P and Q disjoint?):

$P \cap Q = {6}$ The sets are not disjoint since they share the element 6.

Example 3:

Consider sets: $X = {a, b, c}$ $Y = {b, c, d}$

a. Union (X ∪ Y): $X \cup Y = {a, b, c, d}$

b. Intersection (X ∩ Y): $X \cap Y = {b, c}$

c. Complement of X (with respect to the universal set U): $U = {a, b, c, d, e}$ $X^{'} = U - X = {d, e}$

Example 4:

Consider sets: $M = {1, 2, 3, 4}$ $N = {3, 4, 5, 6}$

a. Symmetric Difference (M △ N): $M △ N = {1, 2, 5, 6}$

b. Power Set of M (P(M)): $P (M) = {{}, {1}, {2}, {3}, {4}, {1, 2}, {1, 3}, {1, 4}, {2, 3}, {2, 4}, {3, 4}, {1, 2, 3}, {1, 2, 4}, {1, 3, 4}, {2, 3, 4}, {1, 2, 3, 4}}$

Share to Social

BASIC SET THEORY

BASIC SET THEORY

What is a Set?

Types of Sets

Example 1:

Example 2:

Example 3:

Example 4:

Related Papers