Problem 14 - Discrete Mathematics with Applications

Question

Mark each as true or false. $$\mathbf{b} \subseteq\\{\mathbf{a}, \mathbf{b}, \mathbf{c}\\}$$

Step-by-Step Solution

Answer

The statement \(\{\mathbf{b}\} \subseteq \{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\) is true, as every element in the first set is present in the second set.

Step 1: Understanding subsets

A set A is a subset of set B, denoted as \(A \subseteq B\), if every element of A is also an element of B. In other words, if all elements of A can be found in B, then A is a subset of B.

Step 2: Analyzing the given sets

Now let's analyze the given sets: - The first set is \(\{\mathbf{b}\}\), which contains only one element, \(\mathbf{b}\). - The second set is \(\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\), which contains three elements: \(\mathbf{a}\), \(\mathbf{b}\), and \(\mathbf{c}\).

Step 3: Checking if the first set is a subset of the second set

To determine if \(\{\mathbf{b}\} \subseteq \{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\), we need to check if the element \(\mathbf{b}\) from the first set is also present in the second set. We can clearly see that the element \(\mathbf{b}\) is indeed present in the second set, \(\{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\).

Step 4: Conclusion

Since every element in the first set is also present in the second set, the statement \(\{\mathbf{b}\} \subseteq \{\mathbf{a}, \mathbf{b}, \mathbf{c}\}\) is true.

Key Concepts

Set Theory

Set theory is a fundamental part of mathematics that deals with the concept of sets, which are collections of objects. Sets can include numbers, letters, or any elements that share a common property. These elements are usually enclosed in curly brackets like this: \( \{a, b, c\} \).

The primary relationships in set theory include subsets, unions, intersections, and complements. A **subset** is a set where every element is also found in another set. For example, if you have sets \( A = \{1, 2\} \) and \( B = \{1, 2, 3\} \), set \( A \) is a subset of set \( B \), because all elements of \( A \) (which are 1 and 2) are also in \( B \).

Understanding subsets helps in solving many mathematical problems because it involves checking if one set's elements are part of another, providing foundational skills essential for advanced topics in mathematics.

Mathematical Logic

Mathematical logic provides a structure for reasoning through mathematical concepts and problems. It involves using logical structures like propositions, logical connectives, and quantifiers. While often considered abstract, logic is used in various areas of mathematics, including set theory.

In the context of subsets, logic is used to verify statements such as \( A \subseteq B \). This statement is true if every logical element from set \( A \) is also found in set \( B \).

Let's break it down:

• If you claim "set \( \{ b \} \) is a subset of \( \{ a, b, c \} \)," logic dictates you must show \( b \) is included in the second set.
• This creates a true logical statement because no elements are excluded.

Logical thinking helps identify and prove truths within the mathematical framework, making it a critical tool for problem-solving.

Elements of a Set

Every set is composed of elements. The definition and identification of these elements are pivotal in set-related operations. An element can be a number, letter, symbol, or even another set. For example, in the set \( \{x, y, z\} \), each symbol \( x, y, \) and \( z \) represents an element.

Understanding the elements allows us to determine relationships such as subsets. When asked if \( \{b\} \) is a subset of \( \{a, b, c\} \), you need first to identify what constitutes each set:

• Set \( \{b\} \) has one element: \( b \).
• Set \( \{a, b, c\} \) has three elements: \( a, b, \) and \( c \).

After identifying these elements, assessing the subset relationship becomes a straightforward task. Recognizing \( b \) is indeed part of the larger set confirms the subset condition is true, showcasing how elements play a central role in defining set theory concepts.