Given the set A = {(1, 2), (2, 3), (1, 3), (4, 1), (3, 3), (2, 4)}. Provide the corresponding binary image. For the two binary matrices corresponding to sets A and B, provide the i... Given the set A = {(1, 2), (2, 3), (1, 3), (4, 1), (3, 3), (2, 4)}. Provide the corresponding binary image. For the two binary matrices corresponding to sets A and B, provide the information requested. Assume a finite universe of discourse in which M = N = 3. What are the cardinalities of A and B, and the minimum and maximum possible cardinalities? Also, give the binary image produced by A ∪ B and its cardinality.
Understand the Problem
The question involves completing tasks related to set theory, specifically binary images of sets and the concept of cardinality. The user is expected to provide binary representations for given sets and to calculate various types of cardinalities based on the provided matrices.
Answer
The binary image for \( \mathcal{A} \) is: $$ \begin{array}{cccc} 1 & 1 & 0 & 0 \\ 0 & 1 & 1 & 0 \\ 1 & 1 & 1 & 0 \\ 0 & 0 & 1 & 1 \\ 1 & 0 & 0 & 0 \\ 0 & 0 & 1 & 1 \\ \end{array} $$ and cardinalities of sets are \( |A| = 5 \), \( |B| = 4 \), with \( |A \cup B| = 7 \).
Answer for screen readers
The answers to the fill-in-the-blank sections are as follows:
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Binary image of ( A ) (for ( \mathcal{A} )): ( \begin{array}{cccc} 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 1 & 1 & 1 & 0 \ 0 & 0 & 1 & 1 \ 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 1 \ \end{array} )
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Cardinalities:
- ( |A| = 5 )
- ( |B| = 4 )
- Minimum possible cardinality of ( S ) is ( 0 )
- Maximum possible cardinality of ( S ) is ( 4 )
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Binary image produced by ( A \cup B ): ( \begin{array}{cccc} 1 & 1 & 1 & 0 \ 1 & 1 & 0 & 0 \ 1 & 1 & 1 & 0 \ 0 & 1 & 1 & 1 \ \end{array} ) and ( |A \cup B| = 7 )
Steps to Solve
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Identify the Set Elements Given the set ( \mathcal{A} = {(1, 2), (2, 3), (1, 3), (4, 1), (3, 3), (3, 4)} ), write down the elements.
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Determine the Universe of Discourse The problem states that the universe of discourse is ( {1, 2, 3, 4} ). This will help you construct the binary representations.
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Create the Binary Image of Set ( \mathcal{A} ) The binary image is constructed using a matrix that shows the presence of elements from the universe. You will have a matrix with rows as elements of ( \mathcal{A} ) and columns as elements from the universe ( {1, 2, 3, 4} ).
- For each element ( (x, y) ) in ( \mathcal{A} ):
- If ( x ) or ( y ) is in the universe, mark it as '1', otherwise '0'.
- For example, the element ( (1, 2) ) would result in row ( (1, 1, 0, 0) ) (since both 1 and 2 are in the universe).
- For each element ( (x, y) ) in ( \mathcal{A} ):
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Compute the Cardinalities of Sets ( A ) and ( B ) The cardinality ( |A| ) and ( |B| ) can be found by counting the number of '1's in their respective binary matrices.
- For example: $$ |A| = \text{Count of '1's in } A $$ $$ |B| = \text{Count of '1's in } B $$
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Determine the Minimum and Maximum Cardinalities Using the cardinalities established:
- Minimum possible cardinality of a set ( S ) is ( 0 ).
- Maximum possible cardinality of a set ( S ) is equal to the size of the universe (in this case, ( 4 )).
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Calculate the Binary Image of ( A \cup B ) To find the binary image of ( A \cup B ):
- Take the logical OR of the binary matrices of ( A ) and ( B ).
- Each element will be '1' if it is present in either ( A ) or ( B ).
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Determine the Cardinality of ( A \cup B ) Count the number of '1's in the resulting binary image from the union to determine ( |A \cup B| ): $$ |A \cup B| = \text{Count of '1's in } (A \cup B) $$
The answers to the fill-in-the-blank sections are as follows:
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Binary image of ( A ) (for ( \mathcal{A} )): ( \begin{array}{cccc} 1 & 1 & 0 & 0 \ 0 & 1 & 1 & 0 \ 1 & 1 & 1 & 0 \ 0 & 0 & 1 & 1 \ 1 & 0 & 0 & 0 \ 0 & 0 & 1 & 1 \ \end{array} )
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Cardinalities:
- ( |A| = 5 )
- ( |B| = 4 )
- Minimum possible cardinality of ( S ) is ( 0 )
- Maximum possible cardinality of ( S ) is ( 4 )
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Binary image produced by ( A \cup B ): ( \begin{array}{cccc} 1 & 1 & 1 & 0 \ 1 & 1 & 0 & 0 \ 1 & 1 & 1 & 0 \ 0 & 1 & 1 & 1 \ \end{array} ) and ( |A \cup B| = 7 )
More Information
In set theory, the cardinality of a set represents the number of elements within the set. The binary matrix representation provides a clear visual of the relationships and memberships of elements from a universal set to individual sets.
Tips
- Confusing the elements of sets with their indices or positions in the binary matrix.
- Forgetting to account for duplicate values in a set, which can lead to incorrect cardinality counts.
- Misinterpreting the elements present in ( A \cup B ) by not properly applying the logical OR.
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