Problem 36 - Applied Mathematics: For the Managerial, Life, and Social Sciences

Question

Let \(U=\\{1,2,3,4,5,6,7,8,9,10\\}\) \(A=\\{1,3,5,7,9\\}, B=\\{2,4,6,8,10\\}\), and \(C=\\{1,2,4,\), \(5,8,9\\}\). List the elements of each set. a. \(A^{c} \cap\left(B \cap C^{c}\right)\) b. \(\left(A \cup B^{c}\right) \cup\left(B \cap C^{c}\right)\) c. \((A \cup B)^{c} \cap C^{c}\)

Step-by-Step Solution

Answer

The short versions of the answers are: a. \(A^c \cap (B \cap C^c) = \{2,6,10\}\) b. \((A \cup B^c) \cup (B \cap C^c) = \{1,2,3,5,6,7,9,10\}\) c. \(((A \cup B)^c) \cap (C^c) = \emptyset\)

Step 1: List the elements of each set

Since the sets are already given, let's first list them for reference. Universal set U: \(\{1,2,3,4,5,6,7,8,9,10\}\) Set A: \(\{1,3,5,7,9\}\) Set B: \(\{2,4,6,8,10\}\) Set C: \(\{1,2,4,5,8,9\}\)

Step 2: Calculate complements of sets A, B, and C

Always calculate complements relative to the universal set U. Complement of A (\(A^c\)): \(\{x \in U: x \notin A\} = \{2,4,6,8,10\}\) Complement of B (\(B^c\)): \(\{x \in U: x \notin B\} = \{1,3,5,7,9\}\) Complement of C (\(C^c\)): \(\{x \in U: x \notin C\} = \{3,6,7,10\}\) Now that we have complements of each set, we can perform the required set operations easily.

Step 3: Calculate \((A^c) \cap (B \cap C^c)\)

First, find the intersection of B and the complement of C: \(B \cap C^c = \{x \in U: x \in B \text{ and } x \in C^c\} = \{2,6,10\}\) Now, find the intersection of the resulting set and the complement of A: \(A^c \cap (B \cap C^c) = \{x \in U: x \in A^c \text{ and } x \in (B \cap C^c)\} = \{2,6,10\}\)

Step 4: Calculate \((A \cup B^c) \cup (B \cap C^c)\)

First, find the union of A and the complement of B: \(A \cup B^c = \{x \in U: x \in A \text{ or } x \in B^c\} = \{1,3,5,7,9\}\) Then, find the union of the resulting set and the intersection of B and the complement of C (which is already calculated in step 3): \((A \cup B^c) \cup (B \cap C^c) = \{x \in U: x \in (A \cup B^c) \text{ or } x \in (B \cap C^c)\} = \{1,2,3,5,6,7,9,10\}\)

Step 5: Calculate \(((A \cup B)^c) \cap (C^c)\)

First, find the union of A and B: \(A \cup B = \{x \in U: x \in A \text{ or } x \in B\} = \{1,2,3,4,5,6,7,8,9,10\}\) Now, find the complement of the resulting set: \((A \cup B)^c = \{x \in U: x \notin (A \cup B)\} = \emptyset\) (Note that the complement is an empty set since the union of A and B completely covers the universal set U) Finally, find the intersection of the resulting set and the complement of C: \(((A \cup B)^c) \cap (C^c) = \{x \in U: x \in (A \cup B)^c \text{ and } x \in C^c\} = \emptyset \) The solutions are as follows: a. \(A^c \cap (B \cap C^c) = \{2,6,10\}\) b. \((A \cup B^c) \cup (B \cap C^c) = \{1,2,3,5,6,7,9,10\}\) c. \(((A \cup B)^c) \cap (C^c) = \emptyset\)

Key Concepts

Universal Set

When we speak about set theory in mathematics, the term universal set is of fundamental importance. It is denoted as U and contains all the possible elements under consideration for a particular discussion or problem. Think of it as the complete set of all objects or numbers you are studying.

For example, in the exercise provided, the universal set is defined as \(U = \{1,2,3,4,5,6,7,8,9,10\}\), which includes all single-digit positive integers from 1 to 10. This set encompasses all the other subsets in the problem, like \(A\) and \(B\). Every element found in any subset comes from this universal pool of numbers. So, when you're performing operations like finding complements or intersections, they are all relative to this universal set.

Complement of a Set

Moving on to the complement of a set, it's a way of referring to elements that are in the universal set but not in the subset we're considering. With respect to our universal set \(U\), the complement of \(A\text{, denoted as } A^c\text{,}\) would include all the numbers that aren't in \(A\).

To illustrate from the solution, \(A^c = \{2,4,6,8,10\}\) because these numbers are in \(U\) but not in \(A\). Understanding complements is crucial for solving various set problems, such as finding intersections and unions with other sets' complements, because it provides a different perspective of the elements that are being discussed.

Set Operations

Lastly, set operations are procedures that combine, relate, or modify sets in various ways. The main operations include union, intersection, and complement.

Here are some brief explanations:

• Union (denoted by \(\cup\)): This finds all unique elements that exist in either of the two sets being united.
• Intersection (denoted by \(\cap\) ): This finds all elements that appear in both sets.
• Complement: As discussed, this finds all elements not present in the subset but in the universal set.

Using the problem as an example, in step 4, \(\left(A \cup B^c\right) \cup \left(B \cap C^c\right)\) calculates elements that are either in \(A\text{, in the complement of } B\text{, or in both sets } B \(\text{and the complement of } C\)\). Understanding these operations is crucial for algebraic manipulation of sets and to analyze the relations between different sets.