INTERVALOS (DEFINICIÓN, TIPOS Y REPRESENTACIONES)

Introduction to Intervals

Overview of Intervals

• The speaker, Licenciado Bolívar, introduces the topic of intervals as a direct consequence of order axioms and supremum action.

• The real number line is presented as the primary laboratory for representing mathematical objects and performing operations on them.

Preliminary Considerations

• Symbols related to order (greater than, less than, etc.) are directly linked to open and closed intervals; ">" relates to open intervals while "≥" relates to closed intervals.

• Logical connectives "and" (∩) and "or" (∪) correspond to intersection and union operations between intervals.

• Correct reading of interval symbols requires identifying the variable first; for example, read as "x < a" instead of "a > x".

Defining Intervals

Interval Definition

• An interval A is defined as a subset of real numbers such that if two numbers exist in this set, all values between them also belong to it.

Representations of Intervals

• Intervals can be represented in three ways:

• Set notation (conjunctive representation)

• Summary notation

• Geometric representation on the real line.

Types of Intervals

Closed Intervals

• Closed intervals include their endpoints; represented geometrically on the real line with points a and b .

• In set notation, closed intervals are expressed as A = x in mathbbR | a < x ≤ b .

Open Intervals

• Open intervals do not include their endpoints; they are defined by strict inequalities: A = x in mathbbR | a < x < b .

Semi-open or Semi-closed Intervals

• These can have one endpoint included:

• Example: A = x ∈ R | a < x ≤ b , where b is closed and a ) is open.

Infinite Intervals

Representation of Infinite Intervals

• Infinite intervals can be represented with limits approaching infinity:

• For example, an interval from negative infinity up to point a: (-∞, a] .

Complete Real Number Line Representation

Operations with Intervals and Complements

Introduction to Interval Operations

• The discussion begins with defining operations on intervals, specifically focusing on finding the complement of interval I1 intersected with interval Y2.

• The intervals are defined: I1 is from -2 (open) to 0 (closed), while I2 ranges from -1 (closed) to 1 (open).

Graphical Representation of Intervals

• A graphical representation illustrates the closed and open nature of the intervals, highlighting that I1 is open at -2 and closed at 0, while I2 is closed at -1 and open at 1.

• The complement of interval I1 is identified as all real numbers not included in this interval.

Finding Intersections

• The complement of I1 is represented graphically, showing it extends from negative infinity to -2 (open), then from 0 (open) to positive infinity.

• To find the intersection between the complement of I1 and Y2, an analysis reveals that elements common to both sets must be considered.

Analyzing Common Elements

• At point zero, since the complement of Y1 does not include zero but Y2 does, this affects the outcome; thus, the intersection remains open.

• The final result for this intersection shows it includes values between 0 (open) and 1 (open).

Union and Its Complement

• Next steps involve finding the union of intervals Y1 and Y2. This union spans from -2 (open) to 1 (open).

• The complement of this union encompasses all real numbers outside this range: from negative infinity up to -2 (closed), combined with values starting from 1 (closed).

Final Operations: Complementing Interval Y2

• Moving forward, we calculate the complement of interval Y2 which runs from -1 (closed) to 1 (open).

• This leads us into subtracting interval I1 from its complement; visual aids help clarify how these segments interact.

Conclusion on Subtraction Results

• When performing subtraction between these two segments, care must be taken regarding inclusion or exclusion at endpoints.