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Chapter 6: Problem 22

Let \(S=\\{\) Barnsley, Manchester United, Southend, Sheffield United, Liverpool, Maroka Swallows, Witbank Aces, Roval Tigers, Dundee United, Lyon' be a universal set, \(A=\\{\) Southend, Liverpool, Maroka Swallows, Royal Tigers\\}, and \(B=\\{\) Barnsley, Manchester United, Southend\\}. Find the numbers indicated. \(n\left((A \cup B)^{\prime}\right)\)

Short Answer

Expert verified
\(n\left((A \cup B)^{\prime}\right) = 4\)

Step by step solution

01

Understanding the given sets

We are given the universal set \(S\) and two subsets, \(A\) and \(B\): \(S= \\{ \text{Barnsley, Manchester United, Southend, Sheffield United, Liverpool, Maroka Swallows, Witbank Aces, Royal Tigers, Dundee United, Lyon} \\}\) \(A= \\{ \text{Southend, Liverpool, Maroka Swallows, Royal Tigers} \\}\) \(B= \\{ \text{Barnsley, Manchester United, Southend} \\}\)
02

Finding the union of sets A and B

To find \(A \cup B\), we combine the unique elements of both subsets A and B: \(A \cup B = \\{ \text{Southend, Liverpool, Maroka Swallows, Royal Tigers, Barnsley, Manchester United} \\}\)
03

Finding the complement of the union

The complement of a set is the difference between the universal set and the given set, so we need to remove the elements in \((A \cup B)\) from \(S\): \((A \cup B)^{\prime} = S - (A \cup B) = \\{ \text{Sheffield United, Witbank Aces, Dundee United, Lyon} \\}\)
04

Counting the number of elements in the complement

Now that we have the complement of the union of sets A and B, we just need to count the number of elements: There are 4 elements in \((A \cup B)^{\prime}\). So, \(n\left((A \cup B)^{\prime}\right) = 4\).

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Complement of a Set
Understanding the complement of a set is crucial in set theory, a branch of mathematical logic that deals with collections of objects. Given a universal set, which includes all possible elements under consideration, the complement of a particular subset refers to the elements that are not in the subset but are in the universal set.

Imagine you have a box of assorted chocolates. The universal set is all the chocolates in the box. If you pick out all the milk chocolate pieces, then the complement of the set of milk chocolates consists of all other chocolates that are not milk chocolate.

In our exercise, for the universal set S and subset A 鈭� B, the complement (A 鈭� B)' includes all teams in S that are neither in A nor in B. It's like finding which teams are not playing in either group A or B. There were 4 teams left out, so the size of the complement is 4.
Union of Sets
The concept of the union of sets comes into play when combining elements from two or more sets without repetition. The symbol for union is 鈭�. The union of sets is similar to making a guest list for a party by combining your friends and your roommate鈥檚 friends, ensuring no one is invited twice.

In the given exercise, the teams from both groups A and B are combined to form the union A 鈭� B. The union includes any team that is either in A, or B, or both. The key here is to include each team once even if a team appears in both groups. Counting how many teams play in at least one of the two groups gives us the size of A 鈭� B.
Finite Mathematics
Finite mathematics is an area of mathematics that is concerned with finite or countable processes. It deals with objects that can be counted one by one. In contrast to topics like calculus, which deals with continuous processes, finite mathematics covers subjects like set theory, probability, statistics, and finance, which all involve discrete elements.

In the context of our exercise, the universal set S is finite, containing a countable number of soccer teams. Similarly, the subsets A, B, and their union are also finite, and we can determine the number of elements they contain. Finite mathematics includes methods for counting and arranging these finite collections, which is exactly what we did when we calculated n((A 鈭� B)') as 4.

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