Problem 27 - Algebra and Trigonometry

Question

Find the indicated set if $$ A=\\{1,2,3,4,5,6,7\\} \quad B=\\{2,4,6,8\\} \quad C=\\{7,8,9,10\\} $$ $$ \begin{array}{ll}{\text { (a) } A \cup B} & {\text { (b) } A \cap B}\end{array} $$

Step-by-Step Solution

Answer

(a) A ∪ B = {1, 2, 3, 4, 5, 6, 7, 8}; (b) A ∩ B = {2, 4, 6}.

Step 1: Understanding 'Union of Sets'

The 'union' of two sets, denoted by \(A \cup B\), is a set that contains all the elements from both sets, without duplication. Here, we will identify all unique elements that belong to either set \(A\) or set \(B\) or both.

Step 2: List Elements of A and B for Union

List the elements of set \(A\): \{1, 2, 3, 4, 5, 6, 7\} and set \(B\): \{2, 4, 6, 8\}. We merge these sets, ensuring each element is listed only once.

Step 3: Calculate A ∪ B

The union \(A \cup B\) includes all elements from both sets. \(A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\).

Step 4: Understanding 'Intersection of Sets'

The 'intersection' of two sets, denoted by \(A \cap B\), consists of elements that are common to both sets \(A\) and \(B\). Here, we determine which elements appear in both sets.

Step 5: List Common Elements for Intersection

Identify elements that appear in both set \(A\) and \(B\). These are \{2, 4, 6\}.

Step 6: Calculate A ∩ B

The intersection \(A \cap B\) includes only the common elements. \(A \cap B = \{2, 4, 6\}\).

Key Concepts

Union of Sets

The union of sets is an important concept in set theory that combines all unique elements from the sets in question.
When considering the union, you imagine bringing together every element from each set without any repetition. For example, with the sets given,

• Set \(A = \{1, 2, 3, 4, 5, 6, 7\}\)
• Set \(B = \{2, 4, 6, 8\}\)

To find the union of sets \(A\) and \(B\), denoted as \(A \cup B\), you list each element once. This means you start with the full set \(A\), and add in any element from \(B\) that isn't already there. For this exercise:

• The union is \(\{1, 2, 3, 4, 5, 6, 7, 8\}\)

This set represents all numbers found in sets \(A\) and \(B\), combined together.

Intersection of Sets

The intersection of sets is a central operation where you identify elements common to all sets involved. This operation helps in finding shared traits or numbers between sets. In mathematical terms, the intersection gives you elements that appear in every participating set.

For the given sets,

• Set \(A = \{1, 2, 3, 4, 5, 6, 7\}\)
• Set \(B = \{2, 4, 6, 8\}\)

The intersection of \(A\) and \(B\), denoted as \(A \cap B\), involves finding numbers that belong to both sets. By comparing each element, we see:

• The intersection is \(\{2, 4, 6\}\).

These numbers are found in both \(A\) and \(B\), showcasing shared membership.

Mathematical Sets

Mathematical sets are fundamental in understanding how we group numbers and objects based on specific criteria. Sets are essentially collections of distinct elements, such as numbers or objects, that are grouped together based on shared properties.
These collections are crucial in various mathematical and logical operations like union and intersection, which help us analyze and manipulate data.

A set is usually denoted by curly braces \(\{\} \). For instance,

• Set \(A = \{1, 2, 3, 4, 5, 6, 7\}\)
• Set \(B = \{2, 4, 6, 8\}\)

In this context, each element within the braces is considered a member of the set. Sets are versatile and can be finite or infinite, enabling them to represent anything from a simple list of numbers to complex systems.
Understanding sets and their operations, like unions and intersections, provides a foundational skill useful in both theoretical and applied mathematics.