Problem 6 - Algebra for College Students

Question

Let \(A=\\{1,2,3,4,5,6\\}, B=\\{1,3,5\\}, C=\\{1,6\\},\) and \(D=\\{4\\} .\) Find each set. $$ B \cap C $$

Step-by-Step Solution

Answer

\( B \cap C = \{1\} \)

Step 1 - Understand the Problem

The goal is to find the intersection of sets B and C, which means identifying the common elements in both sets.

Step 2 - List the Elements of B

Set B is given as \( B = \{1, 3, 5\} \).

Step 3 - List the Elements of C

Set C is given as \( C = \{1, 6\} \).

Step 4 - Find the Common Elements

Identify elements that are present in both set B and set C. Compare the elements of \( B \) and \( C \).

Step 5 - Write the Intersection

The common element between the two sets is \( 1 \). Therefore, \( B \cap C = \{1\} \).

Key Concepts

Intersection of sets

The intersection of sets is an operation that finds common elements shared by two or more sets. If we have sets B and C, the intersection is denoted as \(B \cap C\). This notation indicates the set of elements that both B and C contain.

The process of finding the intersection involves comparing each element of one set with each element of the other set. Here, it is important to look for elements that exist in both sets.

For example, given \(B = \{1, 3, 5\}\) and \(C = \{1, 6\}\), we compare the elements and see that the number 1 is present in both sets. Therefore, the intersection \(B \cap C\) results in \(\{1\}\).

Understanding intersections is crucial in set theory and helps with various applications in mathematics and computer science.

Common elements

Common elements are the elements that appear in each of the sets being compared. When identifying common elements, you must carefully compare each element of the sets involved.

Let's go through the process with sets B and C to find their common elements:

• Set B: \(\{1, 3, 5\}\)
• Set C: \(\{1, 6\}\)

We start comparing elements from B and C:
- Compare 1 in B with elements in C: 1 is in C.
- Compare 3 in B with elements in C: 3 is not in C.
- Compare 5 in B with elements in C: 5 is not in C.

From this comparison, we see that the only common element between B and C is 1. Thus, the common element set is \(\{1\}\).

This approach helps clarify the concept of common elements: it’s a matter of simple comparison to identify shared items.

Set notation

Set notation is a systematic way of representing sets and their operations. It's essential for clearly communicating ideas in set theory.

Sets are usually denoted with curly brackets \(\{\}\). For example, a set containing elements 1, 2, and 3 is written as \(\{1, 2, 3\}\).
Another important symbol is the intersection \(\cap\). This symbol denotes the common elements between sets. For instance, if we write \(B \cap C\), we mean a set that contains all elements that are in both B and C.

Using set notation, we can easily describe the operations and relationships between sets. For our exercise, the proper notation for the calculated intersection of sets B and C is \(B \cap C = \{1\}\).

Proper use of set notation makes mathematical expressions concise and easier to understand, which is crucial for solving more complex problems in set theory and beyond.