Problem 13 - Discrete Mathematics with Applications

Question

Using the sets \(A=\\{a, b, e, h\\}, B=\\{b, c, e, f, h\\}, C=\\{c, d, f, g\\},\) and \(U=\\{a, \ldots, h\\},\) find the binary representation of each set. \(B^{\prime}\)

Step-by-Step Solution

Answer

The binary representation of set \(B^{\prime} = \{a, d, g\}\) is \(1 0 0 1 0 0 1 0\).

Step 1: Understand the sets and universal set

First, we need to understand the given sets \(A\), \(B\), \(C\) and universal set \(U\). The universal set \(U = \{a, b, c, d, e, f, g, h\}\) contains all possible elements in the given context. The given sets are: - \(A = \{a, b, e, h\}\) - \(B = \{b, c, e, f, h\}\) - \(C = \{c, d, f, g\}\)

Step 2: Determine the complement of set B, denoted by \(B^{\prime}\)

To find the complement of set \(B\), we must find all elements in the universal set \(U\) that are not in set \(B\). So, we look for elements in \(U\) that are not in \(B\). The complement set is then: \(B^{\prime} = \{a, d, g\}\).

Step 3: Represent the elements using binary numbers

Now we need to find the binary representation of the complement set \(B^{\prime}\). To do this, we will represent each element in the universal set as either a 1 (if it is in \(B^{\prime}\)) or a 0 (if not in \(B^{\prime}\)). So we will have a binary number with 8 digits, one for each element in \(U\) ordered as in \(U\), starting with \(a\) and ending with \(h\). For the complement set \(B^{\prime} = \{a, d, g\}\), the binary representation is: \(1 0 0 1 0 0 1 0\), because - The 1st position (element \(a\)) has a "1" since \(a\) is in \(B^{\prime}\) - The 2nd position (element \(b\)) has a "0" since \(b\) is not in \(B^{\prime}\) - The 3rd position (element \(c\)) has a "0" since \(c\) is not in \(B^{\prime}\) - The 4th position (element \(d\)) has a "1" since \(d\) is in \(B^{\prime}\) - The 5th position (element \(e\)) has a "0" since \(e\) is not in \(B^{\prime}\) - The 6th position (element \(f\)) has a "0" since \(f\) is not in \(B^{\prime}\) - The 7th position (element \(g\)) has a "1" since \(g\) is in \(B^{\prime}\) - The 8th position (element \(h\)) has a "0" since \(h\) is not in \(B^{\prime}\). Therefore, the binary representation of set \(B^{\prime}\) is \(1 0 0 1 0 0 1 0\).

Key Concepts

Binary Representation

Binary representation is a way to express information using two symbols: typically 0s and 1s. It is a fundamental concept in computer science and mathematics for representing data in digital form. In the context of set theory, binary representation is used to simplify and visualize the inclusion of elements in certain sets.

Each element in a universal set can be considered either present or absent in a specific set. When representing a set with binary numbers, each element in the universal set corresponds to a position in a binary string.

• If an element is part of the set we are representing, that position gets a "1".
• If an element is not in the set, it gets a "0".

For instance, when discussing the complement of set B, symbolized as \(B^{\prime}\), with the universal set \(U = \{a, b, c, d, e, f, g, h\}\), we have:

• 1 for any element included in \(B^{\prime}\),
• 0 otherwise.

So, the binary representation of \(B^{\prime}\) becomes \(1 0 0 1 0 0 1 0\) because it includes \(\{a, d, g\}\).

Complement of a Set

The complement of a set is a core concept in set theory. Given a universal set \(U\) and a set \(B\), the complement of \(B\), written as \(B^{\prime}\), includes all elements of \(U\) that are not in \(B\). It's like finding what is missing to complete the whole.

To illustrate this using the sets from our exercise:

• The universal set \(U = \{a, b, c, d, e, f, g, h\}\)
• The set \(B = \{b, c, e, f, h\}\)

The complement \(B^{\prime}\) contains those elements in \(U\) that \(B\) does not have: \(\{a, d, g\}\).

Knowing how to find a set's complement helps in various logical operations and can aid in problem-solving by examining parts that are not explicitly evident.

Universal Set

In set theory, a universal set is a comprehensive set that contains all objects and elements under consideration, providing a complete backdrop for any discussion involving subsets. The universal set is typically denoted by \(U\).

It allows us to define other sets more easily and serves as the reference point for complements and other operations. In our given exercise, the universal set is \(U = \{a, b, c, d, e, f, g, h\}\). Every element that we are considering in any subset is part of this set.

• All subsets, like \(A\), \(B\), and \(C\), draw from these elements.
• The concept of the universal set ensures there is a "complete pie" from which slices (subsets) can be made.

By establishing a universal set, we define all possible items in your working space, making it easier to perform logical operations like finding complements or intersections linked to this set.