Problem 41 - Precalculus Mathematics for Calculus

Question

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

Step-by-Step Solution

Answer

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

Step 1: Understand Union of Sets

To find the union of two sets, combine all the unique elements from both sets. Union is denoted by the symbol \(\cup\). So, to calculate \(A \cup C\), include every element that belongs to either set \(A\) or set \(C\), without repeating any items.

Step 2: Calculate A ∪ C

List all elements from set \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and set \(C = \{7, 8, 9, 10\}\). Combine them by listing each unique element only once: \(A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\).

Step 3: Understand Intersection of Sets

To find the intersection of two sets, identify the elements that both sets have in common. Intersection is denoted by the symbol \(\cap\). For \(A \cap C\), you will find elements that appear in both set \(A\) and set \(C\).

Step 4: Calculate A ∩ C

Compare the elements of \(A\) and \(C\) to find common elements. Set \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and set \(C = \{7, 8, 9, 10\}\) share the number \(7\). Therefore, the intersection is \(A \cap C = \{7\}\).

Key Concepts

Union of Sets

The union of sets is a fundamental concept in set theory that involves combining all elements from two or more sets. When you create a union between two sets, say set \( A \) and set \( C \), you are essentially creating a new set that contains all the elements from both sets. The key principle here is that each element is included only once in the resultant set, even if it appears in both sets originally. The symbol used to represent the union is \( \cup \).

• For example, to find \( A \cup C \), take all unique elements from sets \( A = \{1, 2, 3, 4, 5, 6, 7\} \) and \( C = \{7, 8, 9, 10\} \).
• After listing all unique elements, you get \( A \cup C = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\} \).

This process ensures that you have a comprehensive collection of elements from all sets involved, without any repetitions. Understanding unions is crucial when you're trying to gather all possibilities from different groups since it allows for a holistic view of the elements involved.

Intersection of Sets

When discussing the intersection of sets, focus on finding commonality between two or more sets. The intersection involves identifying elements that are present in both sets. This concept is crucial when you want to determine what is shared between groups. The symbol for intersection is \( \cap \).

• Let's consider sets \( A = \{1, 2, 3, 4, 5, 6, 7\} \) and \( C = \{7, 8, 9, 10\} \).
• For \( A \cap C \), examine which elements exist in both sets; in this case, they both contain the number \( 7 \).
• Thus, the intersection is \( A \cap C = \{7\} \).

Intersections highlight shared elements and can be pivotal in situations where finding a common ground or overlap between different sets is necessary. By understanding intersections, you can analyze data for overlaps and shared associations effectively.

Elements and Membership

In set theory, understanding elements and membership is fundamental to analyzing any set. An element refers to an individual object or number contained within a set, and membership denotes whether an element belongs to a particular set.

• To determine the elements of a set, list all individual items that make it up, such as \( A = \{1, 2, 3, 4, 5, 6, 7\} \).
• To check membership, ascertain if a certain element exists within a set. For example, the number \( 4 \) is a member of set \( A \), but it is not a member of set \( C \). Similarly, the number \( 8 \) is a member of set \( C \) but not set \( A \).

Correctly identifying elements and understanding their membership helps you accurately perform operations such as union and intersection. This foundational knowledge is critical for solving problems and understanding relationships between sets in various mathematical contexts.