Problem 35 - Algebra and Trigonometry

Question

\(35-38=\) Find the indicated set if \(A=\\{1,2,3,4,5,6,7\\} \quad B=\\{2,4,6,8\\}\) \(C=\\{7,8,9,10\\}\) $$ \begin{array}{ll}{\text { (a) } A \cup B} & {\text { (b) } A \cap B}\end{array} $$

Step-by-Step Solution

Answer

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

Step 1: Understanding 'Union' of Sets

The union of two sets, denoted as \(A \cup B\), is a set containing all distinct elements from both sets. To find \(A \cup B\), list all unique elements present in either set \(A\) or set \(B\).

Step 2: Solving for \(A \cup B\)

Start by listing elements of set \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and set \(B = \{2, 4, 6, 8\}\). Include each element from both sets, avoiding duplicates. Thus, \(A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\).

Step 3: Understanding 'Intersection' of Sets

The intersection of two sets, denoted as \(A \cap B\), is a set containing only the elements that are present in both sets. To find \(A \cap B\), identify common elements shared by set \(A\) and set \(B\).

Step 4: Solving for \(A \cap B\)

Compare elements in \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and \(B = \{2, 4, 6, 8\}\) and list elements that appear in both sets. Here, the common elements are \{2, 4, 6\}. Thus, \(A \cap B = \{2, 4, 6\}\).

Key Concepts

Union of Sets

The concept of a union of sets is a fundamental operation in set theory. When we talk about the union, we denote it as \(A \cup B\). This operation combines all the elements from both sets \(A\) and \(B\) into a single set. However, in a union, each element appears only once even if it is present in both sets, as we only list unique elements.

To find the union of \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and \(B = \{2, 4, 6, 8\}\), we list elements that appear in either set. This means we go through each element in both sets \(A\) and \(B\), checking for duplicates, and including each number once in our new set.

This operation leads us to \(A \cup B = \{1, 2, 3, 4, 5, 6, 7, 8\}\). It's like combining two circles of a Venn diagram where the entire space covered by both circles is represented.

Intersection of Sets

While the union of sets is about collecting all distinct elements, the intersection of sets, denoted as \(A \cap B\), is about finding common ground. Here, we focus solely on elements present in both sets \(A\) and \(B\).

For set \(A = \{1, 2, 3, 4, 5, 6, 7\}\) and set \(B = \{2, 4, 6, 8\}\), the goal is to list elements common to both. By comparing these two sets, we identify the elements \(2, 4,\) and \(6\), since they appear in both set \(A\) and set \(B\). Hence, the intersection is \(A \cap B = \{2, 4, 6\}\).

Imagine overlapping parts of a Venn diagram; the intersection represents the shared part or the middle area that both circles overlap.

Elements in Sets

Understanding elements in a set is crucial for performing operations like union and intersection. A set is simply a collection of distinct objects, which are called elements. In mathematical sets, elements are typically numbers, symbols, or points that can be identified clearly.

Consider set \(A = \{1, 2, 3, 4, 5, 6, 7\}\), where the numbers \(1\), \(2\), and so on are the elements of the set \(A\). Similarly, set \(B = \{2, 4, 6, 8\}\) has elements \(2\), \(4\), \(6\), and \(8\). It's important to ensure that each element in a set is unique. That's why in the union of sets, duplicates are not listed.

When performing set operations like union \(A \cup B\) or intersection \(A \cap B\), knowing the elements within each set helps in identifying which values to include in your result. Always remember, when dealing with elements in sets, order doesn't matter, but uniqueness does.