Problem 37 - Algebra and Trigonometry

Question

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

Step-by-Step Solution

Answer

(a) \(A \cup C = \{1,2,3,4,5,6,7,8,9,10\}\); (b) \(A \cap C = \{7\}\)

Step 1: Understand the Given Sets

We have three sets given: - Set \( A = \{1,2,3,4,5,6,7\} \)- Set \( B = \{2,4,6,8\} \)- Set \( C = \{7,8,9,10\} \)

Step 2: Calculate the Union of Sets A and C

The union of two sets, denoted by \( A \cup C \), includes all elements that are in either set \( A \) or set \( C \) or in both. Thus:\( A \cup C = \{1,2,3,4,5,6,7\} \cup \{7,8,9,10\} = \{1,2,3,4,5,6,7,8,9,10\} \)

Step 3: Calculate the Intersection of Sets A and C

The intersection of two sets, denoted by \( A \cap C \), includes only the elements that are present in both set \( A \) and set \( C \). Thus:\( A \cap C = \{1,2,3,4,5,6,7\} \cap \{7,8,9,10\} = \{7\} \)

Key Concepts

Union of Sets

In set theory, the concept of union allows us to combine different sets into one. When we say "union of sets," we refer to a set that includes all the elements from the sets being considered. The symbol used for the union is \( \cup \).
For example, given sets \( A = \{1,2,3,4,5,6,7\} \) and \( C = \{7,8,9,10\} \), the union \( A \cup C \) includes each distinct element found in either set or both.

• Start with the first set: \( \{1,2,3,4,5,6,7\} \)
• Then, add any new elements from the second set \( \{7,8,9,10\} \) not already included.

The union, \( A \cup C = \{1,2,3,4,5,6,7,8,9,10\} \), combines all these distinct elements in a single set.
Remember that in union, elements are not repeated. If an element appears in both sets, it appears only once in the resulting union. This is an essential characteristic of the union operation in set theory.

Intersection of Sets

The intersection of sets is a fundamental concept in set theory that identifies common elements shared by two or more sets. When we talk about the intersection, we are focusing on what is common between the sets. The symbol for intersection is \( \cap \).
Let's look at our example with sets \( A = \{1,2,3,4,5,6,7\} \) and \( C = \{7,8,9,10\} \).

• Identify shared elements: The only number present in both sets \( A \) and \( C \) is \( 7 \).

Therefore, \( A \cap C = \{7\} \).
The intersection operation helps to highlight overlapping elements within sets. This can be especially useful when trying to find shared items in various groups or datasets.

Mathematical Sets

In mathematics, sets are one of the most fundamental concepts, serving as a building block for various other ideas and operations within the field.
A set is essentially a collection of distinct objects, considered as an object in its own right. The elements of a set can be anything: numbers, letters, symbols, and even other sets.

Some key properties of mathematical sets include:

• Unordered collection: Sets do not consider the order of elements, so \( \{1, 2\} \) is the same as \( \{2, 1\} \).
• No repeated elements: A set automatically eliminates duplicates. For instance, \( \{1, 1, 2\} \) is simply \( \{1, 2\} \).
• Notation: Sets are usually denoted using curly braces. For example, a set of numbers from 1 to 5 is \( \{1, 2, 3, 4, 5\} \).
• Membership: We use the symbol \( \in \) to denote membership; so, if \( x \) is an element of set \( A \), we write \( x \in A \).

Understanding these basics of mathematical sets can serve as a valuable tool when exploring more advanced mathematical concepts.