Problem 35 - 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. \((A \cap B) \cup C\) b. \((A \cup B \cup C)^{c}\) c. \((A \cap B \cap C)^{c}\)

Step-by-Step Solution

Answer

a. (A ∩ B) ∪ C = \(\{1,2,4,5,8,9\}\) b. (A ∪ B ∪ C)^c = \(\emptyset\) c. (A ∩ B ∩ C)^c = \(\{1,2,3,4,5,6,7,8,9,10\}\)

Step 1: A. Finding (A ∩ B) ∪ C

First, let's find A ∩ B, which denotes the intersection of sets A and B, i.e., the elements common to both A and B. Recall that A = {1,3,5,7,9} and B = {2,4,6,8,10}. There are no common elements in A and B, so A ∩ B = Ø (empty set). Now we find the union of the intersection we just found with set C: (A ∩ B) ∪ C = Ø ∪ C = C Since the union of the empty set with any set is just that set. So, the answer is (A ∩ B) ∪ C = {1,2,4,5,8,9}.

Step 2: B. Finding (A ∪ B ∪ C)^c

First, let's find the union of the three sets A, B, and C: A ∪ B = {1,2,3,4,5,6,7,8,9,10} since A and B partition the universal set U. Now we find (A ∪ B) ∪ C = {1,2,3,4,5,6,7,8,9,10} ∪ {1,2,4,5,8,9} which is still equal to the universal set U. Now, we need to find the complement of this union with respect to the universal set U: (A ∪ B ∪ C)^c = U^c = Ø (empty set) since the union is equal to the universal set.

Step 3: C. Finding (A ∩ B ∩ C)^c

As we found in part A, A ∩ B = Ø (empty set). Now, let's find the intersection of this result with set C: (A ∩ B) ∩ C = Ø ∩ C = Ø (empty set) The intersection of an empty set with any other set results in an empty set. Finally, find the complement of this intersection with respect to the universal set U: (A ∩ B ∩ C)^c = (Ø)^c = U The complement of the empty set with respect to U is the entire universal set. Thus, (A ∩ B ∩ C)^c = {1,2,3,4,5,6,7,8,9,10}.

Key Concepts

Universal Set

In set theory, a universal set is a fundamental concept that encompasses all possible elements for a particular discussion. It serves as the boundary for other sets under consideration. Think of it like a giant container that holds every element being discussed in the context.
The universal set is denoted by the symbol \(U\). In our given problem, the universal set \(U\) includes all integers from 1 to 10: \[ U = \{1,2,3,4,5,6,7,8,9,10\} \].
All other sets, including \(A\), \(B\), and \(C\), are subsets of this universal set. Understanding what falls within \(U\) ensures we correctly interpret operations involving complements.

Intersection of Sets

The intersection of two sets finds common elements shared between them. Using the symbol \(\cap\), we define \(A \cap B\) as elements found in both set \(A\) and set \(B\).
In our case, \(A = \{1,3,5,7,9\}\) and \(B = \{2,4,6,8,10\}\). Upon comparing these, we see there are no common elements, resulting in \(A \cap B = \emptyset\) (an empty set).
This concept is crucial as it directly impacts the solution's next requirement of taking further unions or complements.
If any two sets share no elements, their intersection will always be the empty set.

Union of Sets

The union of sets combines all elements from the given sets, essentially pooling them together. The symbol \(\cup\) represents union, and the operation encompasses every unique element from each involved set.
For instance, given \(A\) and \(B\) with elements as described earlier, \(A \cup B = U\) because \(A\) and \(B\) together represent all elements of the universal set. Similarly, \((A \cup B) \cup C\) also leads to \(U\) as \(C\) introduces no new elements not already within \(U\).
Union is a comprehensive operation: by pooling without repetition, we consider the widest possible scope of elements.

Complement of a Set

The complement of a set refers to the elements not present in that set, when considered within a universal set. Using the symbol \(^c\), if \(X\) is a subset of a universal set \(U\), then \(X^c\) includes everything in \(U\) that is not in \(X\).
For example, in the problem, \((A \cup B \cup C)^c\) represents all elements in \(U\) that are not in \((A \cup B \cup C)\). Since \((A \cup B \cup C)\) equals the entire \(U\), its complement is \(\emptyset\). Understanding complement involves recognizing what the universal set includes, and then identifying what's absent in the subset in question.