In Exercises 107–110, use graphs to find each set. [1,3) ∩ (0,4)

Verified step by step guidance

1. Identify the type of intervals given in the problem. The interval [1,3) is a closed interval at 1 and open at 3, meaning it includes 1 but not 3. The interval (0,4) is open at both 0 and 4, meaning it does not include 0 or 4.
2. Visualize or sketch the intervals on a number line. Place [1,3) and (0,4) on the same number line to see where they overlap.
3. Determine the common elements between the two intervals. The overlap of [1,3) and (0,4) is where you will find the intersection.
4. Identify the type of interval for the intersection based on where the intervals overlap. Since 1 is included in [1,3) and is within the bounds of (0,4), and 3 is not included in either interval, the intersection forms a new interval.
5. Write the interval of the intersection using proper interval notation, considering the inclusivity or exclusivity of the endpoints based on the overlap observed.

Key Concepts

Interval Notation

Interval notation is a mathematical notation used to represent a range of values. It uses brackets and parentheses to indicate whether endpoints are included or excluded. For example, [1, 3) means that 1 is included in the interval, while 3 is not, indicating all numbers from 1 up to but not including 3.

Set Intersection

Set intersection refers to the operation of finding common elements between two or more sets. The intersection of sets A and B, denoted as A ∩ B, includes only those elements that are present in both sets. In the context of intervals, this means identifying the overlapping range of values that satisfy both conditions.

Graphing Intervals

Graphing intervals involves visually representing the range of values on a number line. Each interval is depicted with a line segment, where closed intervals are marked with solid dots and open intervals with open dots. This visual representation helps in easily identifying overlaps and intersections between different intervals.