Problem 24 - 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. $$ (A \oplus B)-C $$

Step-by-Step Solution

Answer

The binary representation of the expression \((A \oplus B) - C\) is \(10000000\), and the resulting set is \(\{a\}\).

Step 1: Recall the concept of binary representation

For each element in the universal set \(U\), if it is present in the set, we put a '1' in its corresponding position; if it is absent, we put a '0'. For example, for the sets \(A=\{a, b, e, h\}\) and \(U=\{a, ..., h\}\), the binary representation of \(A\) would be: $$ A = 11001001 $$

Step 2: Find binary representation of given sets

Using the above method of binary representation: $$ A = 11001001 \\ B = 01001101 \\ C = 00100110 \\ U = 11111111 $$ Note that the binary representation of \(U\) is the set of all elements, hence all its entries are 1's.

Step 3: Perform the XOR operation on sets A and B

XOR operation is performed element-wise on two given sets' binary representation. A bit in the resulting set is '1' if the corresponding bits in the given sets are different; otherwise, it's '0'. Perform the XOR operation on A and B: $$ A \oplus B = 10000100 $$

Step 4: Perform the set difference operation on the result of step 3 and set C

A set difference operation \((A \oplus B) - C\) removes all the elements present in set C from the set \(A \oplus B\). $$ (A \oplus B) = 10000100 $$ $$ C = 00100110 $$ Perform the set difference by using element-wise AND operation on the complement of C with (A ⊕ B). To find the complement of C, change '1' to '0', and '0' to '1': $$ \overline{C} = 11011001 $$ Now, perform the element-wise AND operation between (A ⊕ B) and the complement of C: $$ (A \oplus B) - C = (A \oplus B) \cap \overline{C} = 10000000 $$

Step 5: Interpret the final binary representation as a set

With the resulting binary representation, we can now find what elements from the universal set are present in it: $$ (A \oplus B) - C = 10000000 \Rightarrow \{a\} $$ Therefore, the binary representation of the expression \((A \oplus B) - C\) is 10000000 and the resulting set is \(\{a\}\).

Key Concepts

Set Theory

Set theory is the mathematical study of collections of objects, which are called sets. In set theory, objects within a set are known as elements, and they do not have any particular order. Sets are typically denoted with curly brackets, and the universal set (\(U\)) represents the collection of all possible elements under consideration. For example, the set \(A=\{a, b, e, h\}\) consists of the elements \(a, b, e, \) and \(h\).

In the context of binary representation, sets are symbolized using a string of binary digits (0s and 1s), where each digit represents the presence (1) or absence (0) of an element from the universal set in the given set. As shown in the solution, the set \(A\) corresponds to the binary sequence 11001001, where each '1' indicates that \(A\) contains the corresponding element from \(U\), and each '0' that it does not.

XOR Operation

The XOR operation, also known as the exclusive OR, is a binary operation that takes two bits as input and outputs '1' if and only if exactly one of the input bits is '1'. If both input bits are the same, the output is '0'. This operation is particularly useful in set theory when dealing with binary representations of sets.

The XOR operation can help in identifying elements that are in one set or the other, but not in both. In binary terms, you perform the XOR operation by comparing corresponding bits in the binary representations of two sets, as shown in the exercise with \(A \oplus B\). The result, in this case, is 10000100, indicating that the element represented by the first digit is in one set and not the other, while all other elements are either in both sets or in neither.

Set Difference

The set difference operation is a fundamental concept in set theory, symbolized as \(A - B\) or \(A \backslash B\). This operation results in a new set that consists of all the elements that are in set \(A\) but not in set \(B\). When we apply the set difference to binary representations of sets, we essentially remove all the '1's in \(B\) from \(A\).

Following the given solution, to compute the set difference \(\left(A \oplus B\right) - C\), we first find the binary representation of the XOR operation result, then we negate the binary representation of \(C\) to obtain \(\overline{C}\), which is its complement. We can then perform an AND operation with \(\left(A \oplus B\right)\) and \(\overline{C}\) to get the final binary representation of the set difference. A '1' in this final representation indicates that the element represented is in \(A\) XOR \(B\) but not in \(C\), as exemplified by the result 10000000, which corresponds to the set \(\{a\}\).