Problem 4 - Precalculus

Question

Find the indicated intersection or union and simplify if possible. Express your answers in interval notation. $$ (-\infty, 4] \cap(0, \infty) $$

Step-by-Step Solution

Answer

The intersection is \((0, 4]\).

Step 1: Understanding the Problem

We need to find the intersection of two intervals, \((-\infty, 4]\) and \((0, \infty)\).

Step 2: Identifying the Intervals

The interval \((-\infty, 4]\) includes all numbers less than or equal to 4, while \((0, \infty)\) includes all numbers greater than 0.

Step 3: Finding the Intersection

The intersection of two sets is the set of elements that exist in both. So, we look for numbers that are greater than 0 (from the second interval) and less than or equal to 4 (from the first interval).

Step 4: Expressing the Intersection in Interval Notation

The intersection of \((-\infty, 4]\) and \((0, \infty)\) is the set of numbers that are greater than 0 and less than or equal to 4.This can be expressed in interval notation as: \((0, 4]\).

Step 5: Verification and Simplification

We check the boundary conditions: - The number 0 is not included since the second interval \((0, \infty)\) does not include it.- The number 4 is included as it satisfies \((-\infty, 4]\). The simplified intersection in interval notation is correct.

Key Concepts

Interval Notation

Understanding interval notation is essential in set theory as it provides a compact way to represent sets of numbers. In interval notation, we use brackets and parentheses to denote the boundaries of an interval. These symbols help us identify the inclusion or exclusion of the endpoints in the set.

Here’s a breakdown of what each symbol means:

• "(": This symbol denotes that an endpoint is not included in the interval. For example, \( (a, b) \) means all numbers greater than \( a \) and less than \( b \).
• ")": Similar to "(", indicates the endpoint is not included.
• "[": This bracket indicates that an endpoint is included in the interval. So, \( [a, b] \) includes both \( a \) and \( b \).
• "]": Works like "[", showing inclusion of the boundary number.

In the exercise, the interval \( (-\infty, 4] \) implies all numbers less than or equal to 4, while \( (0, \infty) \) covers all numbers greater than zero. Knowing how these notational elements operate is crucial for solving set problems effectively.

Intersection of Sets

When working with sets, one of the key operations you will encounter is the intersection. The intersection of sets is the collection of elements or numbers that are common to both sets.

To find the intersection, imagine a Venn diagram and look for the overlapping region of the circles representing each set. Only the numbers or elements found in this overlap belong to the intersection.

In mathematical terms, the intersection of two sets \( A \) and \( B \) is denoted as \( A \cap B \).

In our exercise example, we explored the intersection between \( (-\infty, 4] \) and \( (0, \infty) \). The intersection is \( (0, 4] \) because:

• Numbers from 0 to 4 appear in both sets.
• 0 is not included as the second interval \( (0, \infty) \) excludes it.
• 4 is included as it meets the criteria of being in \( (-\infty, 4] \).

Recognizing the intersection's boundaries and confirming the inclusion or exclusion of endpoints is vital for correct interval notation.

Union of Sets

Another fundamental operation in set theory is finding the union of sets. This operation combines all elements from the given sets into one.

The union of sets is essentially a gathering of all unique elements present in either set or both. Unlike intersection, where commonality is required, the union simply gathers all unique points.

In notation, the union of two sets \( A \) and \( B \) is expressed as \( A \cup B \). The resulting set includes every element in \( A \) and \( B \), without duplicates.

Let’s say we were to find the union of \( (-\infty, 4] \) and \( (0, \infty) \). The union would be \( (-\infty, \infty) \), since all numbers would be gathered without repetition:

• This includes any number less than or equal to 4 from \( (-\infty, 4] \).
• Also, any number greater than 0 from \( (0, \infty) \).

By understanding this operation, you can easily visualize and combine sets to form new, comprehensive sets. Just remember, the union is an inclusive operation, pushing the boundaries to cover all numbers from the sets always.