The intervals of the straight line or simply straight line remind us of the concept of continuity (continuous numbers). With natural numbers, for example, we cannot obtain this continuity, which we call discrete numbers. Be careful, there are different types of infinity. In this post, we will cover operations with ranges of real numbers.

Ranges in discrete numbers

Natural, integer or rational numbers are said to be countable or enumerable sets. When an interval is finite, we can also classify it into discrete intervals.

Example 1.

So if an interval fits one of these definitions, we say it is discrete.

Example 1.

Example of discrete intervals

Represent the range of natural numbers between $1$ and $31$.

$$A = \\{1,2,3,4,5,6,7,8,9,10, \dots, 31\\}$$ Example 1.

Note that the interval $A$ is finite and countable. Therefore, it is a discrete interval.

Continuous Intervals

Real numbers are considered continuous, because given any interval, we have infinite and uncountable numbers within this interval.

Example 1.

Example of Continuous Intervals

Probability uses the concept of continuous numbers, using an interval between 0 (zero) and 1 (one).

Example 1.

Interval union

Let the subsets $A, B$ and $C$ be the real numbers, that is, ranges of real numbers. The union of the intervals is given by:

$$C = A\cup B$$

Example 1 for joining ranges

Let the subsets be $A=[1,3],$ $B=[2,5]$ and we have:

$$A\cup B = [1,3] \cup [2,5] = [1,5]$$

Example 2 for joining ranges

Let the subsets be $A=[1,2],$ $B=[2,5]$ and we have:

$$A\cup B = [1,2] \cup [2,5] = [1,5]$$

Example 3 for joining ranges

Let the subsets be $A=[1,2],$ $B=[2,5]$ and we have:

$$A\cup B = [1,2] \cup [4,5]$$

Note that $A\cup B$ is a disjoint range.

Intersection of ranges

Let the subsets $A, B$ and $C$ be the real numbers, that is, ranges of real numbers. The intersection of the intervals is given by:

$$C = A\cap B$$

Example 1 for interval intersection

Let the subsets be $A=[1,3],$ $B=[2,5]$ and we have:

$$A\cap B = [1,3] \cap [2,5] = [2,3]$$

Example 2 for interval intersection

Let the subsets be $A=[1,2],$ $B=[2,5]$ and we have:

$$A\cap B = [1,2] \cap [2,5] = \\{2\\}$$

Example 3 for interval intersection

Let the subsets be $A=[1,2],$ $B=[2,5]$ and we have:

$$A\cap B = [1,2] \cap [4,5]=\emptyset$$

Note that $A\cap B$ is empty.

Interval difference

Let the subsets $A, B$ and $C$ be the real numbers, that is, ranges of real numbers. The difference between the intervals is given by:

$$C = A- B$$

Example 1 for interval difference

Let the subsets be $A=[1,3],$ $B=[2,5]$ and we have:

$$A\cap B = [1,3] - [2,5] = [1,2]$$

These are the concepts of operations for ranges. Wait for the next posts.