Sets

Sets

A set is defined as a collection of objects called elements. It is usually denoted by a capital letter (e.g. A, B, C). A set can be defined by:

• enumerating its elements within {}, e.g. A = {2, 4, 6, 8}, F = {Emma, Anna, Martin}, V = {a, e, i, o, u, y}. Infinite sets can be written as E = {2, 4, 6, 8, ...}.
• describing the characteristic or rule used to select the elements of a set, e.g. V = {x | x is a vowel}, E = {x | x is an even integer} or, more precisely, E = {x | ∃s ∈ ℤ : x = 2s}. This means that for each x belonging to set E, there is an integer s from ℤ such that x is twice s. This notation states that E is the set of even integers. We use the symbols ∃ (there exists) and ∀ (for all).

Subset

If each element of set A is also an element of set B, then set A is a subset of set B. In this text, we use the symbol ⊂ to mean “is a subset of”.

For example, if A = {1, 2} and B = {1, 2, 3}, we write:

A ⊂ B

The same relationship can be expressed in the opposite direction using the symbol ⊃, which means “is a superset of”:

B ⊃ A

If D = {3, 4}, then D is not a subset of B because not every element of D belongs to B.

The set of all subsets of set A is called the power set of A and is denoted by P(A). If A has n elements, the number of elements of its power set, i.e. the number of subsets of A, is:

For example, if A = {1, 2, 3}, then A has three elements. Therefore, the number of its subsets is:

Set symbols

It is important to distinguish between elements and subsets.

The symbol ∈ means “is an element of”. For example, if A = {1, 2, 3}, we write:

3 ∈ A

The same relationship can be written in the opposite direction using ∋, which means “contains as an element”:

A ∋ 3

The symbol ⊂ describes a relationship between sets. For example:

{3} ⊂ A

This means that {3} is a subset of A. We cannot write 3 ⊂ A because 3 is an element, not a set. Similarly, we do not write {3} ∈ A because the element of A is 3, not the set {3}.

The opposite relationship can be written using ⊃:

A ⊃ {3}

Empty and universal set

A set that contains no elements is called an empty set. We write it as {} or ∅. The empty set is a subset of every set, including itself.

The universal set U is a set containing all objects or elements considered in a particular situation. All other sets being considered are subsets of the universal set.

Venn diagram

A Venn diagram is a diagram that shows logical relationships between different sets. Venn diagrams are typically used to illustrate relationships such as the intersection, union, or difference of sets.

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}

Basic set operations

Union

The union of sets A and B is the set consisting of all elements that belong to A, B, or both sets. We use the symbol ∪:

A ∪ B

We define it using mathematical notation:

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∪ B = {1, 2, 3, 4, 5, 6, 7}

Intersection

The intersection of two sets A and B, written as A ∩ B, consists of all elements that belong to both sets. Mathematically, we define it as:

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∩ B = {3, 4}

Two sets are called disjoint if they have no elements in common. Their intersection is the empty set:

A ∩ B = ∅

Set difference

The difference between sets A and B is the set of all elements that belong to A but do not belong to B. We write:

A B = {x | x ∈ A ∧ x ∉ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A B = {1, 2}

Complement

The complement of a set B consists of all elements of the universal set U that do not belong to B. It is therefore the difference between the universal set U and set B. We denote the complement of B by B^'.

B^' = U B

Example:

U = {1, 2, 3, 4, 5, 6, 7}
B = {1, 2, 3, 4}
B ⊂ U
U ⊃ B
B' = U B = {5, 6, 7}

Symmetric difference

The symmetric difference of two sets A and B is the set of all elements that belong to A or B, but not to both sets. In other words, it contains the elements that do not belong to their intersection. The symmetric difference of sets A and B is denoted by A Δ B.

A Δ B = {x | (x ∈ A ∨ x ∈ B) ∧ x ∉ A ∩ B}

It can also be written as:

A Δ B = (A B) ∪ (B A)

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A Δ B = {1, 2, 5, 6, 7}

Sets

A set is defined as a collection of objects called elements. It is usually denoted by a capital letter (e.g. A, B, C). A set can be defined by:

• enumerating its elements within {}, e.g. A = {2, 4, 6, 8}, F = {Emma, Anna, Martin}, V = {a, e, i, o, u, y}. Infinite sets can be written as E = {2, 4, 6, 8, ...}.
• describing the characteristic or rule used to select the elements of a set, e.g. V = {x | x is a vowel}, E = {x | x is an even integer} or, more precisely, E = {x | ∃s ∈ ℤ : x = 2s}. This means that for each x belonging to set E, there is an integer s from ℤ such that x is twice s. This notation states that E is the set of even integers. We use the symbols ∃ (there exists) and ∀ (for all).

Subset

If each element of set A is also an element of set B, then set A is a subset of set B. In this text, we use the symbol ⊂ to mean “is a subset of”.

For example, if A = {1, 2} and B = {1, 2, 3}, we write:

A ⊂ B

The same relationship can be expressed in the opposite direction using the symbol ⊃, which means “is a superset of”:

B ⊃ A

If D = {3, 4}, then D is not a subset of B because not every element of D belongs to B.

The set of all subsets of set A is called the power set of A and is denoted by P(A). If A has n elements, the number of elements of its power set, i.e. the number of subsets of A, is:

For example, if A = {1, 2, 3}, then A has three elements. Therefore, the number of its subsets is:

Set symbols

It is important to distinguish between elements and subsets.

The symbol ∈ means “is an element of”. For example, if A = {1, 2, 3}, we write:

3 ∈ A

The same relationship can be written in the opposite direction using ∋, which means “contains as an element”:

A ∋ 3

The symbol ⊂ describes a relationship between sets. For example:

{3} ⊂ A

This means that {3} is a subset of A. We cannot write 3 ⊂ A because 3 is an element, not a set. Similarly, we do not write {3} ∈ A because the element of A is 3, not the set {3}.

The opposite relationship can be written using ⊃:

A ⊃ {3}

Empty and universal set

A set that contains no elements is called an empty set. We write it as {} or ∅. The empty set is a subset of every set, including itself.

The universal set U is a set containing all objects or elements considered in a particular situation. All other sets being considered are subsets of the universal set.

Venn diagram

A Venn diagram is a diagram that shows logical relationships between different sets. Venn diagrams are typically used to illustrate relationships such as the intersection, union, or difference of sets.

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}

Basic set operations

Union

The union of sets A and B is the set consisting of all elements that belong to A, B, or both sets. We use the symbol ∪:

A ∪ B

We define it using mathematical notation:

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∪ B = {1, 2, 3, 4, 5, 6, 7}

Intersection

The intersection of two sets A and B, written as A ∩ B, consists of all elements that belong to both sets. Mathematically, we define it as:

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∩ B = {3, 4}

Two sets are called disjoint if they have no elements in common. Their intersection is the empty set:

A ∩ B = ∅

Set difference

The difference between sets A and B is the set of all elements that belong to A but do not belong to B. We write:

A B = {x | x ∈ A ∧ x ∉ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A B = {1, 2}

Complement

The complement of a set B consists of all elements of the universal set U that do not belong to B. It is therefore the difference between the universal set U and set B. We denote the complement of B by B^'.

B^' = U B

Example:

U = {1, 2, 3, 4, 5, 6, 7}
B = {1, 2, 3, 4}
B ⊂ U
U ⊃ B
B' = U B = {5, 6, 7}

Symmetric difference

The symmetric difference of two sets A and B is the set of all elements that belong to A or B, but not to both sets. In other words, it contains the elements that do not belong to their intersection. The symmetric difference of sets A and B is denoted by A Δ B.

A Δ B = {x | (x ∈ A ∨ x ∈ B) ∧ x ∉ A ∩ B}

It can also be written as:

A Δ B = (A B) ∪ (B A)

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A Δ B = {1, 2, 5, 6, 7}

Sets

A set is defined as a collection of objects called elements. It is usually denoted by a capital letter (e.g. A, B, C). A set can be defined by:

• enumerating its elements within {}, e.g. A = {2, 4, 6, 8}, F = {Emma, Anna, Martin}, V = {a, e, i, o, u, y}. Infinite sets can be written as E = {2, 4, 6, 8, ...}.
• describing the characteristic or rule used to select the elements of a set, e.g. V = {x | x is a vowel}, E = {x | x is an even integer} or, more precisely, E = {x | ∃s ∈ ℤ : x = 2s}. This means that for each x belonging to set E, there is an integer s from ℤ such that x is twice s. This notation states that E is the set of even integers. We use the symbols ∃ (there exists) and ∀ (for all).

Subset

If each element of set A is also an element of set B, then set A is a subset of set B. In this text, we use the symbol ⊂ to mean “is a subset of”.

For example, if A = {1, 2} and B = {1, 2, 3}, we write:

A ⊂ B

The same relationship can be expressed in the opposite direction using the symbol ⊃, which means “is a superset of”:

B ⊃ A

If D = {3, 4}, then D is not a subset of B because not every element of D belongs to B.

The set of all subsets of set A is called the power set of A and is denoted by P(A). If A has n elements, the number of elements of its power set, i.e. the number of subsets of A, is:

For example, if A = {1, 2, 3}, then A has three elements. Therefore, the number of its subsets is:

Set symbols

It is important to distinguish between elements and subsets.

The symbol ∈ means “is an element of”. For example, if A = {1, 2, 3}, we write:

3 ∈ A

The same relationship can be written in the opposite direction using ∋, which means “contains as an element”:

A ∋ 3

The symbol ⊂ describes a relationship between sets. For example:

{3} ⊂ A

This means that {3} is a subset of A. We cannot write 3 ⊂ A because 3 is an element, not a set. Similarly, we do not write {3} ∈ A because the element of A is 3, not the set {3}.

The opposite relationship can be written using ⊃:

A ⊃ {3}

Empty and universal set

A set that contains no elements is called an empty set. We write it as {} or ∅. The empty set is a subset of every set, including itself.

The universal set U is a set containing all objects or elements considered in a particular situation. All other sets being considered are subsets of the universal set.

Venn diagram

A Venn diagram is a diagram that shows logical relationships between different sets. Venn diagrams are typically used to illustrate relationships such as the intersection, union, or difference of sets.

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}

Basic set operations

Union

The union of sets A and B is the set consisting of all elements that belong to A, B, or both sets. We use the symbol ∪:

A ∪ B

We define it using mathematical notation:

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∪ B = {1, 2, 3, 4, 5, 6, 7}

Intersection

The intersection of two sets A and B, written as A ∩ B, consists of all elements that belong to both sets. Mathematically, we define it as:

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∩ B = {3, 4}

Two sets are called disjoint if they have no elements in common. Their intersection is the empty set:

A ∩ B = ∅

Set difference

The difference between sets A and B is the set of all elements that belong to A but do not belong to B. We write:

A B = {x | x ∈ A ∧ x ∉ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A B = {1, 2}

Complement

The complement of a set B consists of all elements of the universal set U that do not belong to B. It is therefore the difference between the universal set U and set B. We denote the complement of B by B^'.

B^' = U B

Example:

U = {1, 2, 3, 4, 5, 6, 7}
B = {1, 2, 3, 4}
B ⊂ U
U ⊃ B
B' = U B = {5, 6, 7}

Symmetric difference

The symmetric difference of two sets A and B is the set of all elements that belong to A or B, but not to both sets. In other words, it contains the elements that do not belong to their intersection. The symmetric difference of sets A and B is denoted by A Δ B.

A Δ B = {x | (x ∈ A ∨ x ∈ B) ∧ x ∉ A ∩ B}

It can also be written as:

A Δ B = (A B) ∪ (B A)

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A Δ B = {1, 2, 5, 6, 7}

Sets

A set is defined as a collection of objects called elements. It is usually denoted by a capital letter (e.g. A, B, C). A set can be defined by:

• enumerating its elements within {}, e.g. A = {2, 4, 6, 8}, F = {Emma, Anna, Martin}, V = {a, e, i, o, u, y}. Infinite sets can be written as E = {2, 4, 6, 8, ...}.
• describing the characteristic or rule used to select the elements of a set, e.g. V = {x | x is a vowel}, E = {x | x is an even integer} or, more precisely, E = {x | ∃s ∈ ℤ : x = 2s}. This means that for each x belonging to set E, there is an integer s from ℤ such that x is twice s. This notation states that E is the set of even integers. We use the symbols ∃ (there exists) and ∀ (for all).

Subset

If each element of set A is also an element of set B, then set A is a subset of set B. In this text, we use the symbol ⊂ to mean “is a subset of”.

For example, if A = {1, 2} and B = {1, 2, 3}, we write:

A ⊂ B

The same relationship can be expressed in the opposite direction using the symbol ⊃, which means “is a superset of”:

B ⊃ A

If D = {3, 4}, then D is not a subset of B because not every element of D belongs to B.

The set of all subsets of set A is called the power set of A and is denoted by P(A). If A has n elements, the number of elements of its power set, i.e. the number of subsets of A, is:

For example, if A = {1, 2, 3}, then A has three elements. Therefore, the number of its subsets is:

Set symbols

It is important to distinguish between elements and subsets.

The symbol ∈ means “is an element of”. For example, if A = {1, 2, 3}, we write:

3 ∈ A

The same relationship can be written in the opposite direction using ∋, which means “contains as an element”:

A ∋ 3

The symbol ⊂ describes a relationship between sets. For example:

{3} ⊂ A

This means that {3} is a subset of A. We cannot write 3 ⊂ A because 3 is an element, not a set. Similarly, we do not write {3} ∈ A because the element of A is 3, not the set {3}.

The opposite relationship can be written using ⊃:

A ⊃ {3}

Empty and universal set

A set that contains no elements is called an empty set. We write it as {} or ∅. The empty set is a subset of every set, including itself.

The universal set U is a set containing all objects or elements considered in a particular situation. All other sets being considered are subsets of the universal set.

Venn diagram

A Venn diagram is a diagram that shows logical relationships between different sets. Venn diagrams are typically used to illustrate relationships such as the intersection, union, or difference of sets.

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}

Basic set operations

Union

The union of sets A and B is the set consisting of all elements that belong to A, B, or both sets. We use the symbol ∪:

A ∪ B

We define it using mathematical notation:

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∪ B = {1, 2, 3, 4, 5, 6, 7}

Intersection

The intersection of two sets A and B, written as A ∩ B, consists of all elements that belong to both sets. Mathematically, we define it as:

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A ∩ B = {3, 4}

Two sets are called disjoint if they have no elements in common. Their intersection is the empty set:

A ∩ B = ∅

Set difference

The difference between sets A and B is the set of all elements that belong to A but do not belong to B. We write:

A B = {x | x ∈ A ∧ x ∉ B}

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A B = {1, 2}

Complement

The complement of a set B consists of all elements of the universal set U that do not belong to B. It is therefore the difference between the universal set U and set B. We denote the complement of B by B^'.

B^' = U B

Example:

U = {1, 2, 3, 4, 5, 6, 7}
B = {1, 2, 3, 4}
B ⊂ U
U ⊃ B
B' = U B = {5, 6, 7}

Symmetric difference

The symmetric difference of two sets A and B is the set of all elements that belong to A or B, but not to both sets. In other words, it contains the elements that do not belong to their intersection. The symmetric difference of sets A and B is denoted by A Δ B.

A Δ B = {x | (x ∈ A ∨ x ∈ B) ∧ x ∉ A ∩ B}

It can also be written as:

A Δ B = (A B) ∪ (B A)

Example:

A = {1, 2, 3, 4}
B = {3, 4, 5, 6, 7}
A Δ B = {1, 2, 5, 6, 7}

Sets

A set is defined as a set of objects called elements . It is always marked by a capitalized letter (e.g. A, B, C). Can be defined by:

• enumerating terms within {}, e.g. the set A = {2, 4, 6, 8}. F (first names) = {Emma, Anna, Martin}, V (vowels) = {a, e, i, o, u, y}. Infinite sets can be written as A = {2, 4, 6, 8, ...}.
• describing the characteristic / rule used to select a set element, e.g. V = {x: x is a vowel}, A = {x | x is an even number} or another way: A = {x | ∃s∈Z : x= 2s} (for each x belonging to set A there is a whole number s from Z for which x is its double - this is the mathematical notation stating that A is a set of even numbers / numbers divisible by two without a remainder). We use symbols ∃ (exists ...) and ∀ (for all ...)

Equality and size of the sets

Two sets are equal if they have the same elements. The order of the elements does not matter. The sets {4, 5, 6}, {5, 6, 4} and {6, 4, 5} are the same. We are not interested in repeating elements either. The sets {4, 4, 4, 5, 6, 6} and {4, 5, 6} are equal because we are only interested in one instance of each element. < / p>

If the element belongs to the set A, use the symbol ∈. If A = {1, 2, 3}, then 1 ∈ A (we read that 1 is an element of A, 1 is contained in A). For elements that are not contained in a set, we use the symbol ∉, e.g. 4 ∉ A.

The number of elements in a set is sometimes referred to as cardinality and denoted by ||. If C = {0, 1, 2, 3}, then it contains 4 elements: |C| = 4

A set that has a finite number of elements is called finite . A non-finite set is infinite .

Subset

If each element of set A is also included in set B, then set A is a subset of set B. We write as B ⊂ A. If A = {1, 2, 3} and B = {1, 2}, then B ⊂ A. And if D = {3, 4}, then D ⊄ A or {3, 4} ⊄ {1, 2, 3}.

The set of all subsets of set A is called the powerset of set A or also P (A). If A has n elements, then for the number of elements of the powerset (the number of subsets of the set A) is calculated as:

E.g. A = {1, 2, 3} – A has three elements. Then the number of its subsets is: $2^n=2^3=8$

Set symbols

It is necessary to distinguish elements and subsets: symbols ∈ and ⊂ have different meanings. If A={1, 2, 3} we cannot write that 3⊂A (because 3 is an element of A, not a subset) or {3} ∈A (because {3} is a subset, not an element).

Empty and Universal Set

A set that contains no elements is called an empty set . We write it as {} or ∅. An empty set is a subset of each set (including itself).

Universal Set U is a set containing all considered objects or elements. All other sets are subsets of the universal set.

Venn diagram

A Venn Diagram is a diagram that shows all possible logical relationships between different files. Venn diagrams are typically used to illustrate the intersection of sets (elements that are common to multiple sets).

Example:

A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}

Basic set operations

Union

The union A and B is defined as a set consisting of all elements belonging to the set A or B. We use the symbol ∪: A ∪ B

We define it by mathematical notation:

A ∪ B = {x | x ∈ A ∨ x ∈ B}

Example:

A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}
A∪B = {1, 2, 3, 4, 5, 6, 7}

Intersection

The intersection of two sets A and B (written as A ∩ B) consists of all elements that are common to both sets. Mathematically, we define this as:

A ∩ B = {x | x ∈ A ∧ x ∈ B}

Example:

A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}
A ∩ B = {3,4}.

We call two sets as disjoint if they have no common element. Their intersection is an empty set: A ∩ B = ∅.

Set difference

The difference between sets A and B is a set of those elements from set A that are not elements of set B. We denote the difference as:

A - B = A B = x ∈ A ⋀ x ∉ B

Example:

A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}
A - B = {1,2}

Complement

A set complement is a set of all elements that are not in a given set B but are in a set that contains set B as a subset (often using the universal set U). The complement of a set is actually a set difference provided that set B is a subset of another set. We mark B '.

Example:

B = {1,2,3,4}
C = {1, 2, 3, 4, 5, 6, 7}
B ⊂ C
B '= {5, 6, 7}

Symmetrical difference

A symmetrical difference (a symmetrical difference) is a set that contains all elements of both sets that are not in their intersection. The symmetrical difference of the sets A and B is denoted as A Δ B.

A Δ B = {x | x ∈ A ⋀ x ∈ B ⋀ x ∉ A ∩ B}

Example:

A = {1, 2, 3, 4}, B = {3, 4, 5, 6, 7}
A Δ B = {1, 2, 5, 6, 7}