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}
