Problem 18 - 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. \(C \oplus A\)

Step-by-Step Solution

Answer

The binary representation of sets A and C are 11001001 and 00110110, respectively. The symmetric difference of C and A is the universal set U, which has a binary representation of 11111111.

Step 1: Understand set binary representation

Each set's binary representation is a sequence of 1's and 0's, where 1 signifies that an element is in the set, and 0 signifies that it is not. We will represent the elements of \(U\) in alphabetical order, meaning that the first position corresponds to a, the second to b, and so on, until the eighth position corresponds to h.

Step 2: Find the binary representation for set A

Using the representation rule from Step 1, we have: A = {a, b, e, h} The binary representation of A would be 11001001. The first 1 represents the presence of 'a', the second 1 represents 'b', the third 0 represents the absence of 'c', and so on, up to the last 1 representing 'h'.

Step 3: Find the binary representation for set C

Using the same rule as before, the binary representation of set C is: C = {c, d, f, g} The binary representation of C would be 00110110.

Step 4: Compute the symmetric difference of sets C and A

The symmetric difference of two sets, C and A, is defined by \(C \oplus A = (C - A) \cup (A - C)\). In other words, it is the union of elements that are unique to set C and elements that are unique to set A. Compute the symmetric difference: \(C \oplus A = (C - A) \cup (A - C)\) \(= (\\{c, d, f, g\\} - \\{a, b, e, h\\}) \cup (\\{a, b, e, h\\} - \\{c, d, f, g\\})\) \(= \\{c, d, f, g\\} \cup \\{a, b, e, h\\}\) \(= \\{a, b, c, d, e, f, g, h\\}\)

Step 5: Find the binary representation for the result

Since the symmetric difference of C and A turns out to be the universal set U, the binary representation will have all 1's: C \(\oplus\) A = {a, b, c, d, e, f, g, h} The binary representation of the symmetric difference would be 11111111.

Key Concepts

Binary Representation

Binary representation for sets involves a straightforward conversion of set elements into binary code. Consider it similar to creating a checklist of elements in a universal set. Each position in this binary sequence represents an element in the universal set.

• Presence: Denoted by a '1'. If the element is in the set, it gets a 1.
• Absence: Denoted by a '0'. If the element is not part of the set, the spot receives a 0.

When listing the members of the universal set in alphabetical order, we systematically use 1s and 0s to reflect presence or absence. For example, if our universal set is \(U=\{a, b, c, d, e, f, g, h\}\) and we want to represent the set \(A=\{a, b, e, h\}\), we see that 'a', 'b', 'e', and 'h' are present, while 'c', 'd', 'f', and 'g' are not. Hence, the binary representation becomes 11001001.

Set Theory

Set theory forms the basis of understanding collections of objects or numbers, termed as 'sets'. Sets enable mathematicians to systematically study collections and the relationships between them.

• Members: The distinct objects in a set. For example, in set \(A=\{a, b, e, h\}\), the members are 'a', 'b', 'e', and 'h'.
• Operations: Common operations on sets include union, intersection, and difference. Each operation helps in analyzing or manipulating the sets and their elements.

Understanding set theory is essential, as it provides clear rules about how elements combine or relate, facilitating advanced mathematical procedures such as the symmetric difference.

Universal Set

The universal set, often symbolized as \(U\), is the set that contains all possible elements under consideration. It bounds the discussion of sets, defining the "universe" or scope of elements.

• In our example, \(U=\{a, b, c, d, e, f, g, h\}\), universally framing the discussion around these elements.
• Every other set we discuss is seen as a subset of this universal set.

Universal sets help anchor discussions in set theory by providing a complete backdrop, against which the existence or non-existence of each element within other sets is determined.

Symmetric Difference Operation

The symmetric difference is a unique set operation that highlights differences between two sets. It pinpoints elements that exist in either one of the sets, but not in both.

• It is represented by the symbol \( \oplus \).
• For two sets, say \(C\) and \(A\), the symmetric difference \(C \oplus A\) includes elements present in either \(C-A\) or \(A-C\).

The operation is helpful in computing differences clearly, reflecting distinct elements of each set. This ensures a detailed understanding when analyzing such differences within the realm of set theory. The end result becomes a valuable component of problem-solving in mathematics, as illustrated by the exercise where the symmetric difference led to the formation of a fundamental universal set.