Problem 25 - Discrete Mathematics with Applications

Question

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

Step-by-Step Solution

Answer

The binary representation of the given expression \(A \oplus (B \oplus C)\) is 10011011.

Step 1: Calculate the binary representation of sets A, B, and C

To find the binary representation of sets A, B and C, we need to create a binary vector for each set. We first look at the elements of each set and compare them to the universal set U. If the element exists in the set, then put a 1 in the corresponding position, otherwise, put a 0. For set A: a: 1, b: 1, e: 1, h: 1 So, \( A = 11010100 \) For set B: b: 1, c: 1, e: 1, f: 1, h: 1 So, \( B = 01110110 \) For set C: c: 1, d: 1, f: 1, g: 1 So, \( C = 00111010 \)

Step 2: Perform XOR (^) operation on sets B and C

Now we need to perform the XOR operation on sets B and C. We do this by comparing the values of the binary representation of the sets B and C bit by bit. For \( B \oplus C\), we will compare: - 0 ⊕ 0 = 0 - 1 ⊕ 0 = 1 - 1 ⊕ 1 = 0 - 1 ⊕ 1 = 0 - 0 ⊕ 1 = 1 - 1 ⊕ 1 = 0 - 1 ⊕ 0 = 1 - 0 ⊕ 1 = 1 The binary representation of \(B \oplus C\) after XOR operation is: 01001101

Step 3: Perform XOR (^) operation on set A and the result from step 2 (B ⊕ C)

Now we need to perform the XOR operation on set A and the result from step 2 (B ⊕ C). For \( A \oplus (B \oplus C)\), we will compare: - 1 ⊕ 0 = 1 - 1 ⊕ 1 = 0 - 0 ⊕ 0 = 0 - 1 ⊕ 0 = 1 - 0 ⊕ 1 = 1 - 1 ⊕ 1 = 0 - 0 ⊕ 1 = 1 - 0 ⊕ 1 = 1 The binary representation of \(A \oplus (B \oplus C)\) after XOR operation is: 10011011 Now we have found the binary representation of the given expression \(A \oplus (B \oplus C)\), which is 10011011.

Key Concepts

Binary Representation

Binary representation in set operations is a way to describe sets using binary digits (0s and 1s). This method helps us visualize how elements from a set correspond to elements in a universal set. The universal set, in this context, includes all possible elements, denoted here as \( U = \{a, b, c, d, e, f, g, h\} \). Each position in the binary vector corresponds to whether an element from the universal set is present or not.

Here's how you can determine the binary representation for a set:

• List all elements from the universal set.
• For each element, check whether it is present in the given set.
• If the element is present, write 1; if not, write 0.

For example, let's look at set \( A = \{a, b, e, h\} \):

• 'a' is present, so the first digit is 1.
• 'b' is present, so the second digit is 1.
• 'e' is present, so the fifth digit is 1.
• 'h' is present, so the eighth digit is 1.

This results in the binary representation \( 11010100 \). By comparing each element to \( U \), you can now easily create binary vectors for any set.

XOR Operation

The XOR (exclusive or) operation is a bitwise operation used in binary mathematics and set theory. It compares two binary digits and results in a 1 if the digits are different and 0 if they are the same. This operation is particularly useful in comparing sets to find differences.

To illustrate, consider sets \( B \) and \( C \):

• The binary representations are \( B = 01110110 \) and \( C = 00111010 \).
• By performing the XOR operation on each corresponding bit, we determine the difference.
• For example, 0 ⊕ 0 = 0, 1 ⊕ 0 = 1, and 1 ⊕ 1 = 0.

Applying XOR to all bits, \( B \oplus C = 01001101 \). This operation highlights elements that are unique to each set and simplifies finding symmetric differences. In set operations, XOR helps to easily combine and compare binary representations efficiently.

Universal Set

A universal set is a complete set containing all objects or elements under consideration, denoted by \( U \). In problems that involve binary representation, the universal set acts as the basis for comparing other sets.

The universal set ensures:

• Consistency: Every set is measured against the same elements.
• Comprehensiveness: It represents all possible values an element can take.

For example, if \( U = \{a, b, c, d, e, f, g, h\} \), then every subset will be a combination of these elements. The universal set's completeness guarantees no elements are overlooked.
When operating within the universal set, understanding how to convert sets into binary numbers and perform XOR operations becomes straightforward. Recognizing every element’s role in the universal set gives a solid foundation for handling binary representations and set operations.