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 ...)

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}