Problem 24 - College Algebra

Question

Find the intersection of the sets. \(\\{r, e, a, l\\} \cap | l, e, a, r\\}\)

Step-by-Step Solution

Answer

The intersection of the sets \(\{r, e, a, l\}\) and \(\{l, e, a, r\}\) is \(\{r, e, a, l\}\).

Step 1: Understanding Intersection

The intersection of two sets, A and B (notated as \(A \cap B\)), is a set that contains all elements that are in both A and B. This means that to find the intersection of these two sets, we need to find all the elements that they share.

Step 2: Identify Common Elements

The sets are \(\{r, e, a, l\}\) and \(\{l, e, a, r\}\). From visual inspection, we can see that both sets contain the same elements, so the intersection of these two sets includes all of these elements: r, e, a, and l.

Step 3: Write the Intersection Set

Finally, we can write the intersection set using the common elements. The intersection set of the two given sets is thus \(\{r, e, a, l\}\), the same as the given sets in this case, because they contain exactly the same elements.

Key Concepts

Sets

In mathematics, a set is a collection of distinct objects, considered as an object in its own right. Sets can be composed of anything: numbers, symbols, letters, or even other sets. The primary characteristic of a set is that each element appears only once, eliminating any duplicates.

When we talk about a set, we typically enclose the elements in curly braces. For example, the set \( \{1, 2, 3\} \) consists of the numbers 1, 2, and 3. Sets can also be finite or infinite, depending on whether they have a limited number of elements or they continue infinitely.

Some of the basic operations on sets include union, intersection, and difference. Each of these operations plays a vital role in enabling us to manipulate and understand how sets relate to each other in terms of their elements.

Common Elements

The concept of common elements revolves around identifying items that appear in multiple sets. These elements are the basis for determining the intersection of sets.

To find common elements between sets, you compare each item in one set with those in another. The elements that appear in both sets become the common elements. This concept is essential when analyzing intersections, as it helps us determine which elements belong to multiple sets simultaneously.

• If a letter or number is found in both sets, it is considered a common element.
• Determining common elements is straightforward: look at each element individually and see which ones are shared between the sets.

In the example from the exercise, all elements \( \{r, e, a, l\} \) and \( \{l, e, a, r\} \) are common between the sets because both contain exactly the same characters.

Notated As

When working with sets, we use specific notation to describe operations and relationships between sets. One such notation is the intersection, which is depicted by the symbol \(\cap\). The expression \( A \cap B \) represents the intersection of sets A and B.

This notation succinctly indicates that we're interested in the collection of elements shared by both sets. In the exercise, the problem is expressed as \( \{r, e, a, l\} \cap \{l, e, a, r\} \), meaning we need to identify the common elements of these sets.

Such notations are crucial for communicating mathematical ideas clearly and efficiently, particularly when working with complex set operations.

• Intersection \(\cap\): Finds common elements between sets
• Union \(\cup\): Combines all elements from sets (without duplicates)
• Difference \(\setminus\): Elements in one set but not another

Being comfortable with these symbols will help you navigate through various set operations smoothly.