1.1. Sets and their types

Set: A set is a collection of distinct objects, considered as an object in its own right. Sets are fundamental to various areas of mathematics and are used to define nearly everything. Here are some common types of sets:

1. Empty set
2. Singleton set
3. Finite set
4. Infinite set
5. Equal sets
6. Equivalent sets
7. Universal set 
8. Subset
9. Overlapping sets
10. Disjoint sets 
11. Power set 

1.Empty Set: A set with no elements, denoted by $\varnothing$ or $\{\}$. For example, $B=\{\}$ is an empty set.

2.Singleton set: A singleton set (also called a unit set) is a set that contains exactly one element.

For example:

• $A=\{3\}$
• $B=\{\text{“apple”}\}$

In both cases, $A$ and $B$ are singleton sets because they each contain just one element.

3.Finite set: A finite set is a set that contains a specific number of elements, meaning the number of elements (cardinality) in the set is countable and limited.

For example:

• $A=\{1,2,3\}$ is a finite set with three elements.
• $B=\{\text{“red”},\text{“green”},\text{“blue”}\}$ is a finite set with three elements.
• $C=\{\}$ (the empty set) is also considered finite, with 0 elements.

4.Infinite set: An infinite set is a set that has an unlimited or uncountable number of elements. This means the set continues indefinitely and cannot be fully counted. Infinite se2ts can be either countably infinite or uncountably infinite.

5.Equal sets: Two sets $A$ and $B$ are said to be equal if every element of $A$ is also an element of $B$, and every element of $B$ is also an element of $A$. This is written as: $A=B$

Example:

Let:

• $A=\{1,2,3\}$
• $B=\{3,1,2\}$

Even though the elements are listed in a different order, sets $A$ and $B$ are equal because they contain exactly the same elements.

.Equivalent sets:

Two sets $A$ and $B$ are equivalent if there is a one-to-one correspondence (bijection) between their elements. In other words, the sets have the same number of elements but may contain different items. Equivalent sets are denoted as: $A\sim B$

Example:

Let:

• $A=\{1,2,3\}$
• $B=\{\text{“a”},\text{“b”},\text{“c”}\}$

Both sets $A$ and $B$ have three elements, so they are equivalent even though their elements are different.

7.Universal set: A universal set $U$ is the set that includes every object or element related to a particular subject or problem domain. The exact contents of the universal set depend on the context of the problem. For example:

• If you’re working with natural numbers, the universal set might be $U=\mathbb{N}=\{1,2,3,\dots \}$.
• If you’re dealing with letters of the alphabet, the universal set could be $U=\{\text{a, b, c, …, z}\}$.

Let the universal set be $U=\{1,2,3,4,5,6\}$, and consider the set $A=\{1,2,3\}$. In this case:

• $A\subseteq U$
• The complement of $A$, or AcA^cAc, would be Ac={4,5,6}A^c = \{4, 5, 6\}Ac={4,5,6}.

8.Subset: A subset is a set in which all elements are also contained within another set. In other words, set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$. This is written as:

$$A\subseteq B$$

Examples:

• If $A=\{1,2,3\}$ and $B=\{1,2,3,4,5\}$, then $A\subseteq B$ because all elements of $A$ are in $B$.
• If $A=\{1,2,3\}$ and $B=\{1,2,3\}$, then $A\subseteq B$ and $A=B$.

9.Overlapping sets:

Two sets $A$ and $B$ are said to overlap if their intersection is non-empty, i.e., there is at least one element that belongs to both $A$ and $B$. In set notation, this is written as:

A∩B≠∅A \cap B \neq \emptysetA∩B=∅

Example:

Let:

• $A=\{1,2,3,4\}$
• $B=\{3,4,5,6\}$

Here, $A$ and $B$ are overlapping sets because they share the elements $3$ and $4$. The intersection of $A$ and $B$ is:

$$A\cap B=\{3,4\}$$

10.Disjoint sets: Disjoint sets are sets that have no elements in common. In other words, the intersection of two disjoint sets is the empty set.

Definition:

Two sets $A$ and $B$ are called disjoint if:

$$A\cap B=\varnothing$$

This means there is no element that belongs to both $A$ and $B$.

Example:

Let:

• $A=\{1,2,3\}$
• $B=\{4,5,6\}$

Since there are no common elements between $A$ and $B$, they are disjoint sets:

$$A\cap B=\varnothing$$

11.Power set: The power set of a given set is the set of all possible subsets of that set, including the empty set and the set itself. The power set is denoted as $\mathcal{P}(A)$, where $A$ is the original set.

Definition:

For a set $A$, the power set $\mathcal{P}(A)$ is the collection of all subsets of $A$. If $A$ has $n$ elements, then the power set $\mathcal{P}(A)$ will have 2n2^n2n elements.

Example:

Let $A=\{1,2\}$.

The subsets of $A$ are:

1. $\varnothing$ (the empty set)
2. $\{1\}$
3. $\{2\}$
4. $\{1,2\}$ (the set itself)

Therefore, the power set of $A$ is:

$$\mathcal{P}(A)=\{\varnothing ,\{1\},\{2\},\{1,2\}\}$$

1.2. Relation of sets and representation

Relations of sets and their representation are fundamental concepts in set theory. Understanding how sets relate to one another and how these relationships can be represented helps in various mathematical and logical contexts.

1. Relations Between Sets

1. Subset Relation:
   • Set $A$ is a subset of set $B$ if every element of $A$ is also an element of $B$. Denoted as $A\subseteq B$.
   • Example: If $A=\{1,2\}$ and $B=\{1,2,3\}$, then $A\subseteq B$.
2. Proper Subset Relation:
   • Set $A$ is a proper subset of set $B$ if $A\subseteq B$ and A≠BA \neq BA=B. Denoted as $A\subset B$.
   • Example: If $A=\{1,2\}$ and $B=\{1,2,3\}$, then $A\subset B$.
3. Equality:
   • Two sets $A$ and $B$ are equal if they contain exactly the same elements. Denoted as $A=B$.
   • Example: If $A=\{1,2,3\}$ and $B=\{3,1,2\}$, then $A=B$.
4. Intersection:
   • The intersection of sets $A$ and $B$, denoted $A\cap B$, is the set of elements that are in both $A$ and $B$.
   • Example: If $A=\{1,2,3\}$ and $B=\{2,3,4\}$, then $A\cap B=\{2,3\}$.
5. Union:
   • The union of sets $A$ and $B$, denoted $A\cup B$, is the set of all elements that are in either $A$ or $B$ or both.
   • Example: If $A=\{1,2\}$ and $B=\{2,3\}$, then $A\cup B=\{1,2,3\}$.
6. Difference:
   • The difference of sets $A$ and $B$, denoted $A-B$ or $A\setminus B$, is the set of elements that are in $A$ but not in $B$.
   • Example: If $A=\{1,2,3\}$ and $B=\{2,3,4\}$, then $A-B=\{1\}$.
7. Complement:
   • The complement of set $A$, with respect to a universal set $U$, denoted AcA^cAc or $U-A$, is the set of elements in $U$ that are not in $A$.
   • Example: If $U=\{1,2,3,4\}$ and $A=\{1,2\}$, then Ac={3,4}A^c = \{3, 4\}Ac={3,4}.

2. Representation of Sets

1. Roster or Tabular Form:
   • Lists all the elements of the set explicitly.
   • Example: $A=\{1,2,3,4\}$.
2. Set-Builder Notation:
   • Defines a set by a property that its members must satisfy.
   • Example: $B=\{x\mid x\text{ is an even number and }1\le x\le 10\}$.
3. Venn Diagrams:
   • Visual representations showing the relationships between sets. Circles represent sets, and overlapping areas show intersections.
   • Example: Two circles intersecting represent the intersection of two sets, while the area covered by both circles represents their union.
4. Arrow Diagrams:
   • Used in relation to functions or mappings, where elements of one set are paired with elements of another set through arrows.
5. Matrix Representation:
   • Useful for representing relations between sets, especially in graph theory and combinatorics. Elements are arranged in a matrix format where rows and columns represent sets.

1.3Operations on sets with their properties

Operations on sets involve various ways to combine, modify, or compare sets. Understanding these operations and their properties is essential for working with sets in mathematics and related fields.

1. Union of Sets

Definition: The union of two sets $A$ and $B$ is the set of all elements that are in $A$, in $B$, or in both. It is denoted as $A\cup B$.

Properties:

• Commutative: $A\cup B=B\cup A$
• Associative: $(A\cup B)\cup C=A\cup (B\cup C)$
• Idempotent: $A\cup A=A$
• Identity: $A\cup \varnothing =A$

Example: If $A=\{1,2,3\}$ and $B=\{3,4,5\}$, then: $A\cup B=\{1,2,3,4,5\}$

2. Intersection of Sets

Definition: The intersection of two sets $A$ and $B$ is the set of all elements that are in both $A$ and $B$. It is denoted as $A\cap B$.

Properties:

• Commutative: $A\cap B=B\cap A$
• Associative: $(A\cap B)\cap C=A\cap (B\cap C)$
• Idempotent: $A\cap A=A$
• Identity: $A\cap \varnothing =\varnothing$
• Distributive over Union: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$

Example: If $A=\{1,2,3\}$ and $B=\{2,3,4\}$, then: $A\cap B=\{2,3\}$

3. Difference of Sets

Definition: The difference between two sets $A$ and $B$, denoted as $A-B$ or $A\setminus B$, is the set of elements that are in $A$ but not in $B$.

Properties:

• Not Commutative: A−B≠B−AA – B \neq B – AA−B=B−A
• Associative with Respect to the Empty Set: $(A-B)-C=A-(B\cup C)$
• Identity: $A-\varnothing =A$
• Idempotent: $A-A=\varnothing$

Example: If $A=\{1,2,3\}$ and $B=\{2,3,4\}$, then: $A-B=\{1\}$

4. Complement of a Set

Definition: The complement of a set $A$ with respect to a universal set $U$, denoted as AcA^cAc or $U-A$, is the set of all elements in $U$ that are not in $A$.

Properties:

• Complement of Union: (A∪B)c=Ac∩Bc(A \cup B)^c = A^c \cap B^c(A∪B)c=Ac∩Bc
• Complement of Intersection: (A∩B)c=Ac∪Bc(A \cap B)^c = A^c \cup B^c(A∩B)c=Ac∪Bc
• Double Complement: (Ac)c=A(A^c)^c = A(Ac)c=A
• Identity: A∪Ac=UA \cup A^c = UA∪Ac=U and A∩Ac=∅A \cap A^c = \emptysetA∩Ac=∅

Example: Let $U=\{1,2,3,4\}$ and $A=\{1,2\}$. Then: Ac={3,4}A^c = \{3, 4\}Ac={3,4}

5. Symmetric Difference of Sets

Definition: The symmetric difference between two sets $A$ and $B$, denoted as $A\Delta B$, is the set of elements that are in either $A$ or $B$ but not in both. It can be expressed as: $A\Delta B=(A-B)\cup (B-A)$

Properties:

• Commutative: $A\Delta B=B\Delta A$
• Associative: $(A\Delta B)\Delta C=A\Delta (B\Delta C)$
• Identity: $A\Delta \varnothing =A$
• Complement Relation: (AΔB)c=(Ac∩B)∪(A∩Bc)(A \Delta B)^c = (A^c \cap B) \cup (A \cap B^c)(AΔB)c=(Ac∩B)∪(A∩Bc)

Example: If $A=\{1,2,3\}$ and $B=\{2,3,4\}$, then: $A\Delta B=\{1,4\}$

1.4. Cardinal number of sets

The cardinal number (or simply cardinality) of a set is a measure of the “number of elements” in the set. It provides a way to quantify the size of the set.

1. Finite Sets

For a finite set, the cardinality is simply the number of elements in the set.

Definition: The cardinal number of a finite set $A$, denoted $|A|$, is the total count of distinct elements in $A$.

Example:

• If $A=\{1,2,3\}$, then $|A|=3$.

2. Infinite Sets

For infinite sets, the concept of cardinality extends to measure the size of infinitely large sets. Different types of infinite sets can have different cardinalities.

Types of Infinite Cardinalities:

1. Countably Infinite: A set is countably infinite if its elements can be put into one-to-one correspondence with the natural numbers $\mathbb{N}$. The cardinality of a countably infinite set is denoted ℵ0\aleph_0ℵ0​ (aleph-null).

   Example:

   • The set of natural numbers $\mathbb{N}=\{1,2,3,\dots \}$ is countably infinite, so its cardinality is ℵ0\aleph_0ℵ0​.
2. Uncountably Infinite: A set is uncountably infinite if it is not possible to list its elements in a sequence such that each element is paired with a natural number. The cardinality of the set of real numbers $\mathbb{R}$ is denoted $\mathfrak{c}$ (the cardinality of the continuum), and it is strictly larger than ℵ0\aleph_0ℵ0​.

   Example:

   • The set of real numbers $\mathbb{R}$ has a cardinality of $\mathfrak{c}$.

3. Cardinality of Power Sets

The cardinality of the power set of a set $A$ is related to the cardinality of $A$. If $A$ has cardinality $\kappa$, then the power set $\mathcal{P}(A)$ has cardinality 2κ2^\kappa2κ.

Example:

• If $A=\{1,2\}$, then $|A|=2$ and the power set $\mathcal{P}(A)$ has cardinality 22=42^2 = 422=4.

4. Comparing Cardinalities

• Finite Sets: Cardinalities can be directly compared by counting elements.
• Infinite Sets: Cardinalities can be compared using concepts like bijections (one-to-one correspondences) and the hierarchy of infinities. For example, the set of natural numbers and the set of integers both have the same cardinality ℵ0\aleph_0ℵ0​, but the set of real numbers has a larger cardinality.

1.5. Algebra of sets

Algebra of sets refers to the study of operations and properties of sets using algebraic principles. It provides a framework for manipulating and combining sets in a systematic way. Key operations and properties in the algebra of sets include:

1. Set Operations

1. Union ($\cup$):
   • Definition: The union of sets $A$ and $B$ is the set of elements that are in $A$, $B$, or both.
   • Property: $A\cup B=B\cup A$ (Commutative), $(A\cup B)\cup C=A\cup (B\cup C)$ (Associative), $A\cup \varnothing =A$ (Identity).
2. Intersection ($\cap$):
   • Definition: The intersection of sets $A$ and $B$ is the set of elements that are in both $A$ and $B$.
   • Property: $A\cap B=B\cap A$ (Commutative), $(A\cap B)\cap C=A\cap (B\cap C)$ (Associative), $A\cap \varnothing =\varnothing$ (Identity).
3. Difference ($\setminus$):
   • Definition: The difference between sets $A$ and $B$ is the set of elements that are in $A$ but not in $B$.
   • Property: A∖B=A∩BcA \setminus B = A \cap B^cA∖B=A∩Bc, $A\setminus \varnothing =A$ (Identity), $A\setminus A=\varnothing$ (Idempotent).
4. Symmetric Difference ($\Delta$):
   • Definition: The symmetric difference of sets $A$ and $B$ is the set of elements that are in either $A$ or $B$ but not in both.
   • Property: $A\Delta B=(A\setminus B)\cup (B\setminus A)$, $A\Delta B=B\Delta A$ (Commutative), $(A\Delta B)\Delta C=A\Delta (B\Delta C)$ (Associative).
5. Complement (AcA^cAc):
   • Definition: The complement of a set $A$ with respect to a universal set $U$ is the set of all elements in $U$ that are not in $A$.
   • Property: (Ac)c=A(A^c)^c = A(Ac)c=A (Double Complement), A∪Ac=UA \cup A^c = UA∪Ac=U (Complement Law), A∩Ac=∅A \cap A^c = \emptysetA∩Ac=∅ (Complement Law).

2. Laws of Set Algebra

1. Commutative Laws:
   • Union: $A\cup B=B\cup A$
   • Intersection: $A\cap B=B\cap A$
2. Associative Laws:
   • Union: $(A\cup B)\cup C=A\cup (B\cup C)$
   • Intersection: $(A\cap B)\cap C=A\cap (B\cap C)$
3. Distributive Laws:
   • Union over Intersection: $A\cup (B\cap C)=(A\cup B)\cap (A\cup C)$
   • Intersection over Union: $A\cap (B\cup C)=(A\cap B)\cup (A\cap C)$
4. Identity Laws:
   • Union: $A\cup \varnothing =A$
   • Intersection: $A\cap U=A$
5. Complement Laws:
   • Double Complement: (Ac)c=A(A^c)^c = A(Ac)c=A
   • Union with Complement: A∪Ac=UA \cup A^c = UA∪Ac=U
   • Intersection with Complement: A∩Ac=∅A \cap A^c = \emptysetA∩Ac=∅
6. Absorption Laws:
   • Union: $A\cup (A\cap B)=A$
   • Intersection: $A\cap (A\cup B)=A$

3. De Morgan’s Laws

De Morgan’s laws relate the complement of unions and intersections:

1. First Law:

   (A∪B)c=Ac∩Bc(A \cup B)^c = A^c \cap B^c(A∪B)c=Ac∩Bc

2. Second Law:

   (A∩B)c=Ac∪Bc(A \cap B)^c = A^c \cup B^c(A∩B)c=Ac∪Bc

1.6. Euler – Venn diagram of sets

Euler-Venn diagrams are graphical tools used to represent sets and their relationships visually. They help in understanding operations between sets, such as unions, intersections, and differences. While the term “Euler diagram” is sometimes used interchangeably with “Venn diagram,” they have distinct differences:

1. Euler Diagrams

Euler diagrams are used to show the relationships between sets, typically including subsets and disjoint sets. They use circles or other shapes to represent sets, with their spatial arrangements indicating set relationships.

Features:

• Overlapping Circles: Circles may overlap to show common elements.
• No Fixed Number of Sets: Euler diagrams can represent any number of sets, and the number of shapes is not fixed.
• Simplified Representation: They often omit empty or irrelevant parts for clarity.

Example: A diagram with three overlapping circles representing sets $A$, $B$, and $C$, with overlaps indicating shared elements.

2. Venn Diagrams

Venn diagrams are a specific type of Euler diagram used to show all possible logical relations between a finite number of sets. They are especially useful for illustrating set operations and relationships in a clear and comprehensive way.

Features:

• Fixed Number of Circles: For $n$ sets, a Venn diagram uses $n$ circles, where all possible intersections are shown.
• Complete Representation: Venn diagrams include all possible intersections and unions of the sets.
• Visual Representation of Operations: Useful for demonstrating union, intersection, difference, and complement.

Example: A Venn diagram for two sets, $A$ and $B$, shows:

• The area where $A$ and $B$ overlap (intersection).
• The area where only $A$ is present.
• The area where only $B$ is present.
• The area outside both $A$ and $B$ (if the universal set is represented).