Problem 5 - Algebra for College Students

Question

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

Step-by-Step Solution

Answer

\(A \cap D = \{4\}\)

Step 1: Identify Elements of Set A

Set A is given as \( \{1, 2, 3, 4, 5, 6\} \).

Step 2: Identify Elements of Set D

Set D is given as \( \{4\} \).

Step 3: Determine Common Elements

The intersection of two sets, represented by \( \cap \), includes only the elements that are present in both sets. Here, find the elements that \(A\) and \(D\) have in common.

Step 4: Find Common Elements

Looking at sets A and D: \(A = \{1, 2, 3, 4, 5, 6\}\) and \(D = \{4\}\). The only element that is in both sets is 4.

Key Concepts

set theory

Set theory is a fundamental part of mathematics that deals with collections of objects. These collections are called sets. Sets are defined by their elements, and the elements of a set can be anything: numbers, people, letters, or even other sets.
For example, consider the set of numbers from 1 to 6: \(A = \{1, 2, 3, 4, 5, 6\}\).
Sets are usually represented using curly braces and the elements are separated by commas. Knowing basic set theory helps us solve problems related to collections of items and how they interact with each other.

intersection

The intersection of two sets is a fundamental operation in set theory. It results in a new set that contains only the elements that are present in both of the original sets. The symbol for intersection is \(\cap\)\.
For instance, if we have sets \(A = \{1, 2, 3, 4, 5, 6\}\) and \(D = \{4\}\), their intersection is found by identifying which elements are in both sets. In this case, the only common element is \(4\). Thus, the intersection \(A \cap D = \{4\}\).

common elements among sets

Finding common elements among sets is a key part of the intersection operation. It's about spotting which members or elements appear in all of the sets you're comparing.
If we look at our example, we have sets \(A\) and \(D\): \(A = \{1, 2, 3, 4, 5, 6\}\) and \(D = \{4\}\). We need to check each element of set \(D\) against the elements of set \(A\) to see if there is any overlap. Here, you'll find that \(4\) is the only element that appears in both sets.

mathematical notation

Mathematical notation is a system of symbols used to write mathematical concepts concisely and precisely. Understanding these symbols is crucial for solving mathematical problems efficiently.
In our exercise, we encounter the symbols \(\{\}\) (curly braces) which define a set, as well as \(\cap\) which denotes the intersection of two sets. For example:

• The set \(A\) is represented as \(\{1, 2, 3, 4, 5, 6\}\).
• The set \(D\) is \(\{4\}\).
• The intersection is written as \(A \cap D = \{4\}\).

Becoming familiar with this notation will help you read and write mathematical ideas clearly.