Problem 28 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understanding Union of Sets

The union of two sets \( B \) and \( C \) is a set that contains all the elements that are in either \( B \) or \( C \), or in both. We denote this operation as \( B \cup C \). First, list all elements of \( B \) and \( C \): \( B = \{2, 4, 6, 8\} \) and \( C = \{7, 8, 9, 10\} \).

Step 2: Finding Elements of B ∪ C

Combine all unique elements from sets \( B \) and \( C \). Both sets together include: \( \{2, 4, 6, 8, 7, 9, 10\} \). Remove duplicates if any; in this case, 8 appears in both, so list it only once.

Step 3: Understanding Intersection of Sets

The intersection of two sets \( B \) and \( C \) is a set that contains only the elements that are present in both \( B \) and \( C \). We denote this as \( B \cap C \).

Step 4: Finding Elements of B ∩ C

Identify the common elements in sets \( B \) and \( C \). Look for elements that appear in both sets. \( B = \{2, 4, 6, 8\} \) and \( C = \{7, 8, 9, 10\} \), so the common element is \(\{8\}\) since it is the only one shared.

Key Concepts

Understanding Union of Sets

When we talk about the union of sets, we mean combining all their elements to create a new set. This new set includes every item from both sets but does not repeat any item.
The union is represented by the symbol \( \cup \).
To perform the union of sets \( B \) and \( C \), listed as \( B = \{2, 4, 6, 8\} \) and \( C = \{7, 8, 9, 10\} \), you would:

• List out all elements from set \( B \).
• List out all elements from set \( C \).
• Combine these lists into a single set, \( B \cup C = \{2, 4, 6, 8, 7, 9, 10\} \).

Any duplicates, such as \( 8 \) in our example, should only be included once in the final set.
The union set will always be as large as or larger than the largest individual set, and includes all elements found in either set.

Intersection of Sets Explained

The intersection of sets refers to finding elements that two or more sets share. It pinpoints only those items appearing in all sets involved.
This operation is symbolized by \( \cap \).
When calculating the intersection between sets \( B \) and \( C \) — where \( B = \{2, 4, 6, 8\} \) and \( C = \{7, 8, 9, 10\} \) — you proceed as follows:

• Identify all elements in common between the sets.
• In our example, the number \( 8 \) is present in both sets.
• Thus, the intersection, \( B \cap C \), is \( \{8\} \).

The result of an intersection can have fewer or even no elements if there's no overlap, leading to an empty set \( \{\} \).
Remember, only overlapping elements get included.

Elements of a Set: A Basic Understanding

When discussing sets, it's important to understand the idea of elements. An element is simply an individual item contained within a set.

Sets themselves are often represented by curly braces \( \{ \} \), and elements within these braces are separated by commas. For instance, in the set \( A = \{1, 2, 3, 4, 5, 6, 7\} \), the numbers \( 1, 2, 3, 4, 5, 6, \) and \( 7 \) are all considered elements of set \( A \).

Key characteristics of a set's elements:

• They are unique: no element repeats within the same set.
• Order does not matter: \( \{1, 2, 3\} \) is the same set as \( \{3, 2, 1\} \).

Understanding these basics about set elements will assist you in grasping more complex operations like union and intersection.