To find the number of integers within a set, you need to understand the nature of the set's boundaries, particularly if it's an interval. Here's how to approach different types of intervals:
Understanding Intervals
An interval represents a range of numbers. It can be closed (including endpoints), open (excluding endpoints), or half-open (including one endpoint but not the other). The key is to determine which integers fall within those boundaries.
Calculating the Number of Integers in Different Interval Types
Here's a breakdown of how to calculate the number of integers based on the interval type:
-
Closed Interval [a, b]: This interval includes both 'a' and 'b'. The number of integers is calculated as
b - a + 1.- Example: In the interval [2, 6], the integers are 2, 3, 4, 5, and 6. The number of integers is 6 - 2 + 1 = 5.
-
Open Interval (a, b): This interval excludes both 'a' and 'b'. The number of integers is calculated as
b - a - 1.- Example: In the interval (2, 6), the integers are 3, 4, and 5. The number of integers is 6 - 2 - 1 = 3.
-
Half-Open Interval [a, b) or (a, b]: These intervals include one endpoint but not the other. The number of integers is calculated as
b - a.-
Example 1: In the interval [2, 6), the integers are 2, 3, 4, and 5. The number of integers is 6 - 2 = 4.
-
Example 2: In the interval (2, 6], the integers are 3, 4, 5, and 6. The number of integers is 6 - 2 = 4.
-
General Sets
If the set is not an interval, but a list of numbers, simply count the number of integers present in the set.
- Example: The set {1.5, 2, 3, 4.7, 5, 6.2} contains the integers 2, 3, and 5. The number of integers in the set is 3.
Summary
To determine the number of integers in a set, identify whether the set is a defined interval (closed, open, or half-open) and apply the corresponding formula. If it's a general set, simply count the integers.
