Problem 4 - Algebra and Trigonometry

Question

Explain the difference between the following two sets of numbers: $$ A=[-2,5] \quad B=(-2,5) $$

Step-by-Step Solution

Answer

Set A includes the endpoints -2 and 5, while Set B does not include these endpoints.

Step 1: Understand Set A

The set \( A = [-2,5] \) is defined using square brackets. This means \( A \) includes all numbers from -2 to 5, inclusive. This can be represented as \( A = \{ x \,|\, -2 \leq x \leq 5 \} \). Therefore, the numbers -2 and 5 are both part of the set.

Step 2: Understand Set B

The set \( B = (-2,5) \) is defined using parentheses. This indicates that \( B \) includes all numbers between -2 and 5, but does not include the endpoints. This can be expressed as \( B = \{ x \,|\, -2 < x < 5 \} \). Hence, neither -2 nor 5 are included in the set.

Step 3: Identify the Difference

The main difference between sets \( A \) and \( B \) is whether the endpoints -2 and 5 are included. Set \( A \) includes the endpoints, as shown by the square brackets, whereas set \( B \) does not include the endpoints, indicated by the parentheses.

Key Concepts

Inclusive and Exclusive Intervals

Intervals are a way to express a set of numbers between two endpoints, but it's crucial to understand whether those endpoints are included or excluded. This idea is captured by the terms "inclusive" and "exclusive." Inclusive intervals are denoted by square brackets, such as \([-2, 5]\). In this case, the endpoints -2 and 5 are part of the set. It means if you list all numbers in the set, both -2 and 5 will be included. On the other hand, exclusive intervals use parentheses, like \((-2, 5)\), indicating the endpoints are not part of the set. This means the numbers -2 and 5 are not included. So, when you use exclusive interval notation, the set contains numbers strictly between the two values, but not the values themselves.

Endpoints in Intervals

Endpoints mark the start and end of an interval. How these endpoints are handled depends on the symbols used in interval notation. For example, in the set \([-2, 5]\), the endpoints -2 and 5 are part of the set itself. That's because square brackets signify that those numbers are included in the range. But if you see \((-2, 5)\), the endpoints -2 and 5 are excluded. Parentheses are your cue that these values are not members of the set. These tiny symbols in interval notation—square brackets and parentheses—play a significant role in how we interpret mathematical sets. Understanding whether endpoints are part of an interval or not helps us determine the elements that belong in a particular set. It's critical when solving problems in calculus or algebra.

Set Notation

Set notation is a powerful tool to define a collection of elements, often numbers, based on specific criteria. It frequently accompanies interval notation, offering a clear representation of which numbers are part of a set. Take, for example, the inclusive interval set \([-2, 5]\). In set notation, it translates to \(\{ x \mid -2 \leq x \leq 5 \}\), which means all numbers \( x \) satisfying the inequality are included. In contrast, the exclusive interval \((-2, 5)\) is represented as \(\{ x \mid -2 < x < 5 \}\). This notation states that the set contains all values \( x \) strictly between -2 and 5, without including -2 and 5 themselves.Understanding set notation helps build a foundation for more advanced mathematical concepts, making it easier to express and communicate ideas in mathematics and beyond.