Problem 21 - Algebra and Trigonometry

Question

Find the intersection of the sets. $$[1,2,3,4] \cap\\{2,4,5\\}$$

Step-by-Step Solution

Answer

\([2, 4]\)

Step 1: Identification of the sets

Identify the two sets. Set A = \([1,2,3,4]\), Set B = \({2,4,5}\)

Step 2: Find common elements

Inspect both sets A and B for common elements. It is clear that the numbers 2 and 4 appear in both sets A and B.

Step 3: Define the intersection

The intersection of sets A and B is the set of all elements which are common to both A and B. Thus, the intersection of sets A and B is the set that contains the numbers 2 and 4.

Key Concepts

Set Theory

Set theory is a branch of mathematics that deals with the study of collections, known as sets. A set is essentially a collection of distinct objects, considered as an object in its own right. Sets are fundamental to nearly every area of mathematics, serving as the building blocks for more complex mathematical structures. Here are a few essential points to understand about set theory:

• Sets are usually denoted by curly braces, such as \(\{1, 2, 3\}\).
• Elements within a set are written inside the curly braces and separated by commas.
• An element can either belong to a set or it does not; membership is a yes or no question.

Set theory uses various operations, like union, intersection, and difference, to create new sets from existing ones. These operations help in understanding the relationships between different sets. Understanding set theory is crucial for grasping more complex mathematical concepts. It lays the groundwork for areas such as logic, probability, and statistics.

Common Elements

When dealing with sets, a common task is to identify elements that are shared between them, often described as common elements. Common elements are simply the items that appear in more than one set. Recognizing these shared elements is essential for understanding relationships between different data groups.

• A practical approach is to list the items in each set and look for overlaps.
• These overlaps indicate the common elements, such as the numbers 2 and 4 in the sets \([1, 2, 3, 4]\) and \(\{2, 4, 5\}\).

By identifying common elements, we gain insights into the degree of similarity between sets. This concept arises frequently in real-world applications, such as comparing datasets, tracking shared characteristics, or consolidating information from multiple sources. Recognizing common elements leads directly to understanding intersections of sets.

Intersection of Sets

The intersection of sets refers to a new set that contains all the elements that are present in each of the original sets. The symbol used to represent the intersection is \(\cap\). When two sets share elements, those elements become part of the intersection set.To find the intersection of sets, follow these steps:

• List out all the elements in each set.
• Identify elements that appear in both sets.
• Form a new set from these common elements.

For example, if we have Set A = \([1, 2, 3, 4]\) and Set B = \(\{2, 4, 5\}\), the intersection of these sets is \(\{2, 4\}\). This is because 2 and 4 are the only numbers present in both sets. The notion of intersection is essential in mathematics because it helps in narrowing down possibilities or solutions by focusing on commonality rather than differences. Whether in data analysis, probability, or logic, understanding how intersections work is fundamental.