A set is a collection of objects, states, letters, numbers, or any entities. One way to define a set is to assign it a name and list its members (or elements) enclosed in curly brackets { }. For instance, a set named ( A ) can be defined as follows:
– ( A = {a, b, c} )
This notation provides a clear representation of the elements belonging to the set.
In this context, the set A consists of three elements. Any set that has a finite number of elements is called a finite set. The number of elements in a finite set is known as its cardinality, denoted by |S|.
For example, the cardinality of the set A defined above is |A| = 3.
The set of integers, which contains an infinite number of elements, is represented by the symbol Z and can be expressed as:
– Z = {…, -3, -2, -1, 0, 1, 2, 3, …}.
Another way to describe a set is by using descriptive language. For instance, the set N can be defined as “the set of natural numbers, consisting of 0, 1, 2, …”. The symbols ∈ and ∉ denote “is an element of” and “is not an element of,” respectively.
For example:
- 3 ∈ Z (read as “3 is an element of the set Z”),
- 2/3 ∉ Z (read as “2/3 is not an element of the set Z”).
It is also possible to define a set by specifying a rule that identifies all its elements. Consider the following notation:
– A = {n : n ∈ Z and n ≥ 10}.
This expression can be read as:
“A is the set of all n in Z such that n is greater than or equal to 10.”
Therefore, A would be:
– A = {10, 11, 12, 13, …}.
Let’s discuss the clusters that need to be known below.
- B = {0,1}: The set consisting of binary digits.
- N: The set of natural numbers, which includes all positive integers starting from zero.
- Z: The set of integers, which comprises negative integers, zero, and positive integers.
- Q: The set of rational numbers.
- C: The set of complex numbers.
- R: The set of real numbers.
The set of real numbers R includes integers (Z), rational numbers (Q), and irrational numbers (such as π and √2). However, when dealing with quantum computations, we often need to consider a number set larger than R. This necessity introduces the set of complex numbers C, which consists of numbers with both real and imaginary components.
When discussing sets, the symbol “\” is often used to denote the exclusion of an element from a set. For example, R∖{0} represents the set of all real numbers except 0.
The most inclusive set in which operations are performed and that encompasses all other sets is called the Universal Set, denoted by the symbol E. A set with no elements is called the Empty Set, represented by ∅ or {{}.
We should also mention the concept of a Field. A field is an algebraic structure defined by specific rules where addition, subtraction, multiplication, and division by non-zero elements can be performed. The main purpose of a field is to provide a framework where these operations are defined and valid according to certain axioms. The sets of real numbers R and complex numbers C are examples of fields.
A binary operation is a rule that takes two elements from a set and produces another element within the same set. Familiar binary operations include addition and multiplication in the set of real numbers, but there are many other examples that will be discussed in detail later.
Now, let’s discuss properties specific to sets. When performing operations on sets, terms like Intersection, Union, and Complement are commonly used. Below, we will explore these concepts.