Problem 8 - The Calculus with Analytic Geometry

Question

In Exercises 1 through 10, list the elements of the given set if \(A=\\{0,2,4,6,8\\}, B=\\{1,2,4,8\\}, C=\\{1,3,5,7,9\\}\), and \(D=\) \(\\{0,3,6,9\\}\). $$ A \cap C $$

Step-by-Step Solution

Answer

\(\emptyset\)

Step 1: Understand the set notation

The symbol \(\backslashcap\) denotes the intersection of two sets, which means we need to find the common elements between set A and set C.

Step 2: List the elements of set A

The elements of set A are: \(\{0,2,4,6,8\}\).

Step 3: List the elements of set C

The elements of set C are: \(\{1,3,5,7,9\}\).

Step 4: Find the common elements

Compare the elements in sets A and C to identify the elements that are present in both sets. In this case, there are no elements that are present in both A and C.

Step 5: Write down the intersection

Since there are no common elements between sets A and C, the intersection \(A \cap C\) is an empty set: \(\{\}\) or \(\emptyset\).

Key Concepts

Intersection of Sets

In set theory, the intersection of sets is a fundamental concept. The intersection, denoted by the symbol \(\backslashcap\), refers to the set containing all elements that are common to two or more sets.
For example: If we have set A = \{1, 2, 3, 4\} and set B =\{2, 4, 6, 8\}, then the intersection A \cap B =\{2, 4\}.
This is because elements 2 and 4 are present in both sets A and B.
To better understand intersections:

• List down all elements in each set.
• Identify the elements that appear in all sets.
• The resulting set is the intersection.

In our original exercise, sets A and C have no common elements, so their intersection is the empty set.

Empty Set

An empty set, denoted by the symbol \(\backslashemptyset\) or \{\}, is a set that contains no elements.
It is a unique set in set theory and serves as the foundation for constructing other sets.
It's important to note:

• Every set has the empty set as a subset.
• The intersection of two sets with no common elements is the empty set.

In the original exercise, when sets A and C are intersected, the result is an empty set \(\backslashemptyset\) because there are no elements shared between these sets.

Set Notation

Set notation is a mathematical language used to describe collections of objects or numbers. It uses curly braces \{\} to list elements of a set.
For example, \{1, 2, 3\} is a set containing the numbers 1, 2, and 3.
Key elements of set notation include:

• \textbf{Element}: Objects or numbers within a set, such as 1 in \{1, 2, 3\}.
• \textbf{Subset}: A set whose elements are all contained within another set, such as \{1, 2\} being a subset of \{1, 2, 3\}.
• \textbf{Union}: The combination of all elements in two sets, denoted by \(\backslashcup\).
• \textbf{Intersection}: As discussed, the common elements between sets, denoted by \(\backslashcap\).

Understanding set notation is crucial for solving problems in calculus and algebra. It simplifies the representation of complex mathematical ideas.