Problem 19 - Elementary and Intermediate Algebra

Question

Let \(A=\\{0,1,2,3,4,5,6\\}, B=\\{4,6,8,10\\}\) \(C=\\{-3,-1,0,1,2\\},\) and \(D=\\{-3,1,2,5,8\\}\) Find each set. $$ C \cap D $$

Step-by-Step Solution

Answer

The intersection is \( C \cap D = \{-3, 1, 2\} \).

Step 1: Understand the Intersection

To find the intersection of two sets, look for elements that are common to both sets. The intersection is denoted by the symbol \( \cap \). So for this problem, we need to identify the elements present in both sets \( C \) and \( D \).

Step 2: List the Elements of Set C

Set \( C \) is given as \( \{-3,-1,0,1,2\} \). List these elements to determine which ones might also be in set \( D \).

Step 3: List the Elements of Set D

Set \( D \) is given as \( \{-3,1,2,5,8\} \). List these elements to understand which ones could be common with set \( C \).

Step 4: Find Common Elements

Compare the elements from sets \( C \) and \( D \). Identify the numbers present in both sets. From \( C \), the elements \(-3, 1, \) and \( 2 \) are also present in \( D \).

Step 5: Write the Intersection

List the common elements identified in Step 4 to write the intersection. Therefore, \( C \cap D = \{-3, 1, 2\} \).

Key Concepts

Common Elements

When you're comparing two sets, finding the common elements is a key activity. Common elements are numbers or objects that belong to both sets. These are the pieces that connect or overlap the properties or characteristics of the two collections you're analyzing.

To identify common elements, simply list out all the members of each set and see which items appear in both lists. In mathematical terms, these shared elements belong to the intersection of the two sets. This intersection is indicated with the symbol \(\cap\).

Consider sets \( C = \{-3, -1, 0, 1, 2\} \) and \( D = \{-3, 1, 2, 5, 8\} \). Here, \(-3\), \(1\), and \(2\) are found in both \( C \) and \( D \). Therefore, the common elements and the intersection are \( \{-3, 1, 2\} \). Identifying these shared pieces helps to solve problems where understanding overlap is key.

Understanding Sets

Sets are fundamental building blocks in mathematics, representing collections of distinct objects or elements. Comprehending sets is crucial for grasping more advanced mathematical concepts.

There are a few important aspects to know about sets, such as:

• Elements: The distinct objects or numbers in a set. Each element is included only once, even if it appears multiple times in reality.
• Notation: Sets are typically listed with curly braces like \( \{ \} \).
• Empty Set: Also known as the null set, it contains no elements and is represented as \( \emptyset \) or \( \{\} \).

Understanding these features allows you to engage with sets more deeply, using them to analyze data and solve various mathematical problems. In exercises like identifying intersections, knowing each set's elements allows you to find commonalities effectively.

Mathematical Notation

Mathematical notation is the language that we use to express mathematical concepts concisely. It offers clarity and precision, which are crucial in solving problems and communicating results.

For intersections, we rely on specific symbols and structures:

• Symbol for Intersection \( \cap \): Denotes the set of elements common to all involved sets.
• Braces \( \{ \} \): Used to list the elements of a set.
• Comma Separator: Within braces, elements are separated by commas.

For example, to find the intersection \( C \cap D \), we check which elements are common to both \( C = \{-3, -1, 0, 1, 2\} \) and \( D = \{-3, 1, 2, 5, 8\} \), resulting in \( \{-3, 1, 2\} \).

By familiarizing yourself with these symbols, you can efficiently interpret and convey mathematical ideas, making problem-solving more intuitive and structured.