#UNI01: Sets, special numerical sets, and real intervals

Sets can be used as a way of avoiding rounding or calculation errors when trying to address the preciseness of a real value, given there are infinite real numbers in between any two real numbers. When specifying a range, one is accounting that the solution might not be precise, but it’s bounded between two endpoints.

Table of Contents

1. What are sets?
2. Special numerical Sets
3. Set properties
4. Venn and Euler diagrams
5. Set operations (or algebra of sets)
6. Set-builder notation and identities
7. Functions and set properties
8. Real intervals
9. Sources

What are sets?

Sets are a way of grouping different elements that match a premise.

For expressing this, the roster notation is used, where elements are separated by a comma in between curly braces, usually named with an uppercase letter of the alphabet. For example:

[math]A = \{1,2,3,4\}[/math]

Would be the “A” set containing four elements: the 1, the 2, the 3, and the 4.

As per the definition of set, elements can be anything, so the following set is a valid one also.

[math] B = \{blue, red, green\}[/math]

We can abbreviate sets with an ellipsis, whether the set is infinite, or whether we won’t list all the elements

• Infinite:
  • [math] A = \{…3,4,5,6,7….\}[/math]
  • [math] A = \{…3,4,5,6,7\}[/math]
  • [math] A = \{3,4,5,6,7…\}[/math]
• Finite but abbreviated:
  • [math]A=\{1,2,…,10\}[/math]

The notation where we list the elements can be the extensional definition (not abbreviated, listing them all), or ostensive definition (abbreviated, using ellipsis). The third way of defining a set is by it’s semantic definition, where a rule or premise is specified. All elements that make this premise true will be elements of the set. For example:

“Let A be the set of colors in the argentinian flag” would result in the set: [math]A = \{lightblue,\ white,\ yellow\}[/math]

Sets have different properties and operations:

• Cardinality: the amount of elements of a set, noted with: [math]|A|[/math]
  • [math]A=\{1,2\} \rightarrow |A| = 2[/math]
• Membership
• The empty set
• The singleton set
• Arithmetical operations
• Set builder notation

But for now, let’s focus on Special umerical Sets

Special numerical Sets

Special numerical sets are no more no less than sets of numbers. A specific list of frequently referred set numbers (so called ‘special sets of numbers’) defines the following sets:

• Natural numbers [math]{N}[/math]
  • Sometimes called ‘counting numbers’, which can be used, as the name implies, for counting elements. Whether the 0 is included in this set or not is still for debate. For explicitly including or excluding the zero, the following notations are used (correspondently)
    • [math]{N}_{0}[/math]
    • [math]{N}_{-\{0\}}[/math]
• Integers [math]{Z}[/math]
  • This set contains the counting numbers with its negative counterparts, including 0
  • Ex: [math]1,-5,8,2,3,-1,-6,-9999,0[/math]
• Rational numbers [math]{Q}[/math]
  • Rational numbers are numbers that can be expressed as the quotient of two integers (a numerator, and a denominator)
  • Ex: [math]\frac{3}{4},\frac{8}{3}[/math]
• Real numbers [math]{R}[/math]
  • A real number is a number that can be used to measure a one-dimensional quantity (distance, duration, temperature, etc.). All real numbers can be represented can be almost uniquely represented by an infinite decimal representation.
  • Ex: [math]\sqrt{2}, \pi[/math]
• Complex numbers [math]{C}[/math]
  • Complex number are represented with a sum of two compounds: the real part and the imaginary part, like so, where [math]a[/math] and [math]b[/math] are real numbers, and [math]i[/math] is the imaginary unit.
    • [math]a + b*i[/math]
    • The [math]i[/math] element, called the imaginary unit satisfies the equation:
      • [math]i^2 = -1[/math]

Set properties

• Membership
  • Membership of an element in a set is denoted with the [math]\in[/math] (where the element is part of the set) or [math]\notin[/math] (where the element is not part of the set)
  • For example:
    • [math] A = \{ [/math]the first three multiples of 5[math]\}[/math]
    • [math] 5 \in A [/math], [math]10 \in A[/math], [math]15 \in A[/math]
• Empty set
  • Empty sets are sets without elements, denoted as: [math]\{\emptyset\}[/math]
• Singleton set
  • Sets of one element
• Subset
  • When all the elements of set [math]A[/math] are present in set [math]B[/math], we can say that:
    • [math]A[/math] is a subset of [math]B[/math]:
      • [math] A \subset B[/math]
    • [math]A[/math] is contained in [math]B[/math]:
      • [math] A \subset B[/math]
    • [math]B[/math] contains [math]A[/math]:
      • [math]B ⊃ A [/math]

Venn and Euler diagrams

Both Euler and Venn diagrams are ways of representing sets, their memberships and relationships. The difference is that with Euler’s, only possible relationships are represented, while on Venn’s all relationships are represented, even non-possible ones.

Set operations (or algebra of sets)

Sets, as mathematical elements, can be used to perform algebraic operations with. They share operations and identities with the propositional logic and boolean algebra.

• Compliment
  • The complement of [math]A[/math] is the set of all elements that do not belong to [math]A[/math]:
    • [math]A^\complement[/math] or [math]A'[/math]

• Union
  • Union of [math]A[/math] with [math]B[/math] is a resulting set of all elements contained in [math]A[/math], all elements contained in [math]B[/math], or elements contained in both
  • [math]R = A \bigcup B = B \bigcup A[/math]

• Intersection
  • Is the resulting set of all elements contained in [math]A[/math] and contained [math]B[math].
  • [math]R = A \bigcap B = B \bigcap A[/math]
  • If [math] A \cap B = \emptyset[/math], then [math]A[/math] and [math]B[/math] are disjoint.

• Difference
  • The different between [math]A[/math] and [math]B[/math] is a resulting set of all elements contained in [math]A[/math] but not in [math]B[/math]
  • [math] A – B = A \backslash B = A \bigcap B^\complement[/math]

• Symmetric difference
  • The symmetric difference between [math]A[/math] and [math]B[/math] is a resulting set of elements contained in [math]A[/math] or [math]B[/math], but not both
  • [math]A \bigtriangleup B = (A \backslash B) \bigcup (B \backslash A)[/math]
• Cartesian product
  • The set of all order pairs [math](a,b)[/math] where [math]a[/math] is an element of [math]A[/math] and [math]b[/math] is an element of [math]B[/math]
  • [math]R = A \times B [/math]

Set-builder notation and identities

Set-builder notation is the way of defining a set by enumerating its elements, or by stating the properties that its members must satisfy, a predicate. Meaning, a logical formula that evaluates to true for all elements in the set. It can be written by separating the variable (or variables) and the predicate with a bar separator or colon.

[math] A=\{ x | \Phi(x)\}[/math] or [math]\{ x : \Phi(x)\}[/math]

The bar separator or colon can be read as “such that”, “for which” or similar. It indicates that [math]x[/math] will be an element of the set if the predicate evaluates to true for it. In other words, it can be read as “[math]A[/math] contains every [math]x[/math] for which [math]\Phi(x)[/math] is true.

A third element can be added into the notation, which is the domain specification, which means that the elements of the set [math]A[/math] must not only evaluate the predicate to true, but also exist in the specified domain.

[math] A = \{ x \in E : \Phi(x)\}[/math]

Predicates can be a single proposition or a conjunction of them as logical operations:

[math]A = \{ x \in E : \Phi_{0}(x) \wedge \Phi_{1}(x)\}[/math]

And as an extension of the set-builder notation, instead of only specifying a variable (or variables), one can specify an expression:

[math]A = \{ 2x + 1 : x \in {Z} \}[/math] (All odd integers)

Functions and set properties

Sets are very handful when working with functions, their notation allows a very descriptive and precise definition of functions’ domain and image, while also being able to establish relationships between functions’ elements that can be useful to explain and graph the intersection between different functions.

There are three properties related on how the sets of a function (image, domain) correlate. A function can be one of:

• Injective
  • It means that the function maps a distinct element of the domain to only one element of the image. Or in other words, if every element of the codomain is the image of at most one element of the domain.
• Surjective
  • A function where its image is the same as its codomain, for every [math]y[/math] exists at least one [math]x[/math]
• Bijective
  • An injective and surjective function. A one-to-one relation between the domain, the image and the codomain. Exactly one element of the domain is mapped to an element of the codomain.

Real intervals

Real intervals are subsets of the Real numbers set. Intervals are bounded by two points in a linear plane, where its bound points can be included or not in the interval. Interval can be boundless at the initial endpoint, the final endpoint, or both, indicating it is an infinite interval. The notion for specifying a real range stands by enclosing the endpoints separated by comma with parenthesis -that indicate the endpoint next to it is included in the range-, or with brackets -that indicate that the endpoint next to it is not part of the range-. Infinite endpoints are always written next to a parenthesis.

A range can be interpreted as the result of a premise defined set, or as another way of specifying the elements of it.

• [math] (a,b) = \{ x \in {R} \; \diagup \; a < x < b \}[/math]
  • Elements [math]a[/math] and [math]b[/math] are not included in this set
• [math] [a,b] = \{x \in {R} \; \diagup \; a \leq x \leq b \}[/math]
  • Elements [math]a[/math] and [math]b[/math] are included in this set
• [math] (a,b] = \{x \ \in {R} \; \diagup \; a < x \leq b \}[/math]
  • Element [math]a[/math] is not included in the set, but element [math]b[/math] is.
• [math] [a,b) = \{x \in {R} \; \diagup \; a \leq x < b \}[/math]
  • Element [math]b[/math] is not included in the set, but element [math]a[/math] is.
• [math](-\infty, b] = \{x \in {R} \; \diagup \;x \leq b\}[/math]
  • The set contains all elements up until [math]b[/math], including [math]b[/math]
• [math][a, +\infty) = \{x \in {R} \; \diagup \; x \geq a \}[/math]
  • The set contains all elements from until [math]a[/math], including [math]a[/math]

Intervals, like sets, can undergo through arithmetical operations

And similar to sets, intervals can also be related by set operations:

Sources

1. https://en.wikipedia.org/wiki/Set_(mathematics)
2. https://en.wikipedia.org/wiki/Natural_number
3. https://en.wikipedia.org/wiki/Integer
4. https://en.wikipedia.org/wiki/Rational_number
5. https://en.wikipedia.org/wiki/Real_number
6. https://en.wikipedia.org/wiki/Complex_number
7. https://en.wikipedia.org/wiki/Algebra_of_sets
8. https://www.cs.sfu.ca/~ggbaker/zju/math/set-oper.html
9. https://www.math.uwaterloo.ca/~snburris/htdocs/scav/boolean/boolean.html
10. https://en.wikipedia.org/wiki/Set-builder_notation
11. https://en.wikipedia.org/wiki/Euler_diagram
12. https://en.wikipedia.org/wiki/Codomain
13. https://en.wikipedia.org/wiki/Injective_function
14. https://en.wikipedia.org/wiki/Surjective_function
15. https://en.wikipedia.org/wiki/Bijection
16. https://en.wikipedia.org/wiki/Interval_(mathematics)