Question

Let \(U=\\{1,2,3,4,5,6,7,8,9,10\\}\) be a universal set. Let \(A, B, C \subseteq U\) such that \(A=\\{1,3,4,8\\}, B=(2,3,4,5,9,10),\) and \(C=\\{3,5,7,9,10\\},\) Use bit representations for \(A, B,\) and \(C\) together with UNION, INTER, DIFF, and COMP to find the bit representation for the following: (a) \(A \cup B\) (b) \(A \cap B \cap C\) (c) \((A \cup C) \cap B\) (d) \((A-B) \cup C\) (e) \(A \cap(B-(C \cap B))\) (f) \(A-(B-C)\) (g) \((A \cup B) \cup(C-B)\)

Short answer

(a) 1111100111 (b) 0000100000 (c) 0011100011 (d) 1010101111 (e) 0001000000 (f) 1000100100 (g) 1111101111

Step-by-step solution

Convert Sets to Bit Representations

In a universal set with 10 elements, each subset can be represented by a bit string of length 10. A '1' in a position indicates the presence of the corresponding element.- For set \(A = \{1,3,4,8\}\), the bit representation is \(1001100100\).- For set \(B = \{2,3,4,5,9,10\}\), the bit representation is \(0111100011\).- For set \(C = \{3,5,7,9,10\}\), the bit representation is \(0010101011\).

Calculate A ∪ B

The union operation combines elements from both sets, resulting in '1' if either bit is '1':\[A \cup B = 1001100100 \cup 0111100011 = 1111100111\]

Calculate A ∩ B ∩ C

Intersection keeps only elements present in all sets, resulting in '1' if all bits are '1':\[A \cap B \cap C = 1001100100 \cap 0111100011 \cap 0010101011 = 0000100000\]

Calculate (A ∪ C) ∩ B

First, find \(A \cup C\), then intersect with \(B\):\[A \cup C = 1001100100 \cup 0010101011 = 1011101111\]Now, intersect with \(B\):\[(A \cup C) \cap B = 1011101111 \cap 0111100011 = 0011100011\]

Calculate (A - B) ∪ C

First, find \(A - B\) by removing elements from \(A\) present in \(B\), and then take the union with \(C\):\[A - B = 1001100100 - 0111100011 = 1000000100\]Now take the union with \(C\):\[(A - B) \cup C = 1000000100 \cup 0010101011 = 1010101111\]

Calculate A ∩ (B - (C ∩ B))

First, find \(C \cap B\), then \(B - (C \cap B)\), and finally intersect with \(A\):\[C \cap B = 0010101011 \cap 0111100011 = 0010100011\]Subtract from \(B\):\[B - (C \cap B) = 0111100011 - 0010100011 = 0101000000\]Intersect with \(A\):\[A \cap (B - (C \cap B)) = 1001100100 \cap 0101000000 = 0001000000\]

Calculate A - (B - C)

First, find \(B - C\), then subtract from \(A\):\[B - C = 0111100011 - 0010101011 = 0101000000\]Subtract from \(A\):\[A - (B - C) = 1001100100 - 0101000000 = 1000100100\]

Calculate (A ∪ B) ∪ (C - B)

First, find \(C - B\), then union with \(A \cup B\).\[C - B = 0010101011 - 0111100011 = 0000001000\]Now take the union:\[(A \cup B) \cup (C - B) = 1111100111 \cup 0000001000 = 1111101111\]

Key concepts

Bit Representation

Bit representation is a way to express membership of elements in a set using a string of binary digits, or 'bits'. In a universal set with a fixed number of elements, each element can be associated with a position in a bit string. In our example, the universal set has 10 elements, so we use a 10-bit string. Each position in the string corresponds to an element in the universal set. A bit value of '1' at a certain position indicates the element is present in the subset, while a '0' indicates its absence.

For example, for set \(A = \{1, 3, 4, 8\}\), the bit representation is \(1001100100\). This means that elements 1, 3, 4, and 8 are present in set A. Similarly, set \(B = \{2, 3, 4, 5, 9, 10\}\) is represented as \(0111100011\), where elements 2, 3, 4, 5, 9, and 10 are included. This system is highly efficient for performing set operations like union, intersection, and difference in computational environments.

Universal Set

The universal set is the comprehensive set that includes all the elements under consideration, and it serves as a reference for the subsets involved in a problem. In this context, the universal set \(U = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 10\}\) encompasses all elements being discussed.

The role of the universal set is to provide a complete list of possibilities, ensuring subsets like \(A, B,\) and \(C\) are correctly defined within it. This allows operations like set difference and complements to be effectively calculated, as these depend on knowing which elements are not part of a subset. Understanding the universal set is crucial because it dictates the length of the bit representation and helps ensure accurate results in set operations.

Union and Intersection

Union and intersection are fundamental operations in set theory. They allow for combining sets in different ways to analyze and interpret data. The

• Union (\(\cup\)) of two sets includes all elements that are in either set or both. Bitwise, this is represented by a '1' in any position where at least one of the sets has a '1'. For example, if set \(A = \{1, 3, 4, 8\}\) and set \(B = \{2, 3, 4, 5, 9, 10\}\), then \(A \cup B\) results in \(\{1, 2, 3, 4, 5, 8, 9, 10\}\), with bit representation \(1111100111\).
• Intersection (\(\cap\)) only includes elements present in all sets involved in the operation. In bit representation, an intersection results in a '1' only where all sets show '1'. Using set \(A, B,\) and \(C\), where \(C = \{3, 5, 7, 9, 10\}\), the intersection \(A \cap B \cap C\) yields \(\{3\}\), represented as \(0000100000\).

These operations provide powerful tools for analyzing groups of elements, allowing us to see overlaps and combinations efficiently.

Set Difference

Set difference is another key operation that identifies elements present in one set while absent in another. In mathematical terms, the set difference \(A - B\) consists of all elements that are in set \(A\) but not in set \(B\).

In terms of bit representation, this operation is performed by marking '1' only in positions where the first set has a '1' and the second set has a '0'. For example, for \(A = \{1, 3, 4, 8\}\) represented as \(1001100100\) and \(B = \{2, 3, 4, 5, 9, 10\}\) represented as \(0111100011\), the difference \(A - B\) is \(1000000100\), corresponding to the element list \(\{1, 8\}\).

Understanding set difference is beneficial for tasks involving exclusion, such as identifying unique items, and is a common operation in data analysis and computer science.