Question

Let \(A=\\{1,2,3, \ldots, 10\\}, B=12,3,6,8\\},\) and \(C=(3,5,4,8,2\\} .\) Find the following: (a) \(B \cup C\) (b) \(B \cap C\) (c) \(B-C\) (d) \(A-B\) (e) \(A-C\)

Short answer

(a) \( \{12, 3, 6, 8, 5, 4, 2\} \), (b) \( \{3, 8\} \), (c) \( \{12, 6\} \), (d) \( \{1, 4, 5, 7, 9, 10\} \), (e) \( \{1, 6, 7, 9, 10\} \).

Step-by-step solution

Understand Union

The union of two sets, denoted as \( B \cup C \), includes all elements that are in either set or both. To find \( B \cup C \), we take all the unique elements from both sets B and C.

Find B Union C

Set \( B = \{12,3,6,8\} \) and \( C = \{3,5,4,8,2\} \). List all elements and eliminate duplicates: \[ B \cup C = \{12, 3, 6, 8, 5, 4, 2\} \].

Understand Intersection

The intersection of two sets, denoted as \( B \cap C \), includes only the elements that are present in both sets. To find \( B \cap C \), we list the common elements of B and C.

Find B Intersection C

Compare both sets \( B \) and \( C \), then take the common elements: \[ B \cap C = \{3, 8\} \].

Understand Set Difference B-C

The difference of two sets \( B - C \), contains elements that are in B but not in C. Identify elements present in B and not in C for \( B - C \).

Find B-C

Remove elements of C from B: \[ B - C = \{12, 6\} \].

Understand Set Difference A-B

The set difference \( A - B \) involves elements that are in A but not in B. Identify elements in A that do not exist in B.

Find A-B

Set \( A = \{1,2,3, \ldots, 10\} \). Elements in A but not in B are: \[ A - B = \{1, 4, 5, 7, 9, 10\} \].

Understand Set Difference A-C

The set difference \( A - C \) includes elements that are in A but not in C, excluding any common elements.

Find A-C

Remove elements of C from A: \[ A - C = \{1, 6, 7, 9, 10\} \].

Key concepts

Union of Sets

In set theory, the union of two sets is a fundamental operation that combines all elements of the given sets. This is written as \( B \cup C \). To perform a union, you combine every unique element from both sets.

• Example: Consider sets \( B = \{12, 3, 6, 8\} \) and \( C = \{3, 5, 4, 8, 2\} \).
• Identify all elements present in either set.
• Exclude any duplicates when listing the combined set.

Thus, the union \( B \cup C \) results in \( \{12, 3, 6, 8, 5, 4, 2\} \). Union captures the overall diversity of the elements from both sets.

Intersection of Sets

The concept of intersection is crucial when you want to find elements that two sets have in common. This is shown as \( B \cap C \). The intersection identifies shared elements between sets.

• Example: For sets \( B = \{12, 3, 6, 8\} \) and \( C = \{3, 5, 4, 8, 2\} \), we look for common items.
• List elements that appear in both \( B \) and \( C \).

So, \( B \cap C \) gives us \( \{3, 8\} \). Intersection provides insight into the overlap of both sets.

Set Difference

Set difference is about identifying elements in one set that are not in another. It helps in filtering out elements exclusive to a set.

• There are distinct types: \( B - C \) and \( A - B \).
• Example for \( B - C \): Start with set \( B = \{12, 3, 6, 8\} \).
• Remove elements appearing in \( C = \{3, 5, 4, 8, 2\} \).
• The result is \( \{12, 6\} \), the set difference \( B - C \).
• Example for \( A - B \): Consider set \( A = \{1, 2, 3, \ldots, 10\} \).
• Remove elements seen in \( B \) resulting in \( \{1, 4, 5, 7, 9, 10\} \).

Set differences help isolate specific elements, confirming what is exclusive to a particular set.

Discrete Mathematics

Discrete mathematics encompasses various topics, including set theory. It explores structures that are fundamentally distinct or separate. Sets in discrete mathematics are collections of well-defined objects, aiding in the study of discrete elements.

• Elementary Concepts: Include sets, graphs, and functions.
• Real-life Applications: Computer algorithms, data analysis, and cryptography heavily rely on discrete mathematics.

Understanding operations like union, intersection, and set difference can greatly enhance problem-solving skills by structuring information logically. Discrete mathematics is essential in fields that require discrete, organized data, laying the groundwork for comprehension and innovation in modern computing.