Problem 30 - Algebra and Trigonometry

Question

Find the indicated set if $$ A=\\{1,2,3,4,5,6,7\\} \quad B=\\{2,4,6,8\\} \quad C=\\{7,8,9,10\\} $$ $$ \begin{array}{ll}{\text { (a) } A \cup B \cup C} & {\text { (b) } A \cap B \cap C}\end{array} $$

Step-by-Step Solution

Answer

(a) \( A \cup B \cup C = \{1,2,3,4,5,6,7,8,9,10\} \); (b) \( A \cap B \cap C = \emptyset \).

Step 1: Understanding Union of Sets

The union of sets, denoted by \( \cup \), is a set containing all elements from the involved sets, without duplicates. Therefore, to find \( A \cup B \cup C \), list all elements from sets \( A \), \( B \), and \( C \) once.

Step 2: Calculate A ∪ B ∪ C

Combine all elements of sets \( A = \{1,2,3,4,5,6,7\} \), \( B = \{2,4,6,8\} \), and \( C = \{7,8,9,10\} \). The union includes each element only once: \( A \cup B \cup C = \{1,2,3,4,5,6,7,8,9,10\} \).

Step 3: Understanding Intersection of Sets

The intersection of sets, denoted by \( \cap \), is a set containing all elements that are common to all involved sets. Thus, \( A \cap B \cap C \) will include only those elements present in all three sets: \( A \), \( B \), and \( C \).

Step 4: Calculate A ∩ B ∩ C

Identify the common elements in sets \( A = \{1,2,3,4,5,6,7\} \), \( B = \{2,4,6,8\} \), and \( C = \{7,8,9,10\} \). There are no elements common to all three sets. Therefore, \( A \cap B \cap C = \emptyset \).

Key Concepts

Union of Sets

In set theory, the union of sets is represented by the symbol \(\cup\). It signifies a set that contains every distinct element from all the sets involved. When we talk about union, think about gathering all items from each set while ensuring no duplicates. Here, we'll explore how this concept worked in our exercise.

To solve the exercise for the union of sets \(A \cup B \cup C\), take sets \(A = \{1,2,3,4,5,6,7\}\), \(B = \{2,4,6,8\}\), and \(C = \{7,8,9,10\}\). The process involves listing each element from all three sets.

• First, list all elements from \(A\): 1, 2, 3, 4, 5, 6, 7.
• Next, add elements from \(B\) that aren't already included: 8.
• Finally, include elements from \(C\) that haven't yet appeared: 9, 10.

When combined, the union \(A \cup B \cup C = \{1,2,3,4,5,6,7,8,9,10\}\), which is a collection of each unique item from sets \(A\), \(B\), and \(C\). Notice how each number is listed only once, highlighting the distinctiveness rule of union operations.

Intersection of Sets

The intersection of sets is crucial in set theory for understanding shared elements among sets. It's depicted by the symbol \(\cap\) and comprises only those elements present in all the selected sets.

In the given exercise, we were tasked to calculate the intersection of sets \(A \cap B \cap C\), using \(A = \{1,2,3,4,5,6,7\}\), \(B = \{2,4,6,8\}\), and \(C = \{7,8,9,10\}\). Let's see how to find the common elements across these sets.

• First, compare sets \(A\) and \(B\). The commonalities are 2, 4, and 6.
• Next, identify shared elements between \(B\) and \(C\), which gives us only 8.
• Finally, test each common element across all three sets to find the intersection.

For \(A\), \(B\), and \(C\), none of the elements from \(B\) are listed in both other sets \(A\) and \(C\). Hence, \(A \cap B \cap C = \emptyset\) (the empty set). This means no single element exists in all three sets simultaneously.

Elements of Sets

Elements of sets refer to the individual objects or numbers contained within a set. Each item in a set is called an "element," and these elements are crucial in forming the union or intersection of sets.

Let's delve into the basics using our given sets \(A\), \(B\), and \(C\) from the exercise:

• Set \(A\) includes elements: 1, 2, 3, 4, 5, 6, and 7.
• Set \(B\) consists of: 2, 4, 6, and 8.
• Set \(C\) has: 7, 8, 9, and 10.

These elements help us execute operations like union and intersection.
To count as an element of a set, an item must uniquely appear in that collection, indicated by curly braces. When different sets come together (as seen in a union), or overlap (as in an intersection), the involved elements decide the outcome. It’s essential to understand that each element’s presence or absence affects the overall set calculation a lot.