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Give examples of the following, and provide justification as to why your answers are examples of the following. (a) Sets X and Y such that X x Y| = 15. (b) Sets A, B, and C such that ACB, A = C, and |A| < |B| < |C|. (c) Real numbers x and y such that [x+y] = [x+y].

Give examples of the following, and provide justification as to why your answers are examples of the following. (a) Sets X and Y such that X x Y| = 15. (b) Sets A, B, and C such that ACB, A = C, and |A| < |B| < |C|. (c) Real numbers x and y such that [x+y] = [x+y].

Advanced Engineering Mathematics
Advanced Engineering Mathematics
10th Edition
ISBN:9780470458365
Author:Erwin Kreyszig
Publisher:Erwin Kreyszig
Chapter2: Second-order Linear Odes
Section: Chapter Questions
Problem 1RQ
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### Examples and Justifications in Set Theory and Real Numbers #### (a) Cartesian Product of Sets **Problem:** Give examples of sets \( X \) and \( Y \) such that \(|X \times Y| = 15 \). **Solution:** Sets \( X \) and \( Y \) must be chosen such that the product of their cardinalities equals 15. For instance: - Let \( X = \{a, b, c\} \), which has a cardinality of 3. - Let \( Y = \{1, 2, 3, 4, 5\} \), which has a cardinality of 5. Thus, \(|X \times Y| = 3 \times 5 = 15\). **Justification:** The Cartesian product of two sets \(X\) and \(Y\) is the set of all ordered pairs \((x, y)\) where \(x \in X\) and \(y \in Y\). Since \(|X| = 3\) and \(|Y| = 5\), the product \(|X \times Y| = 3 \times 5 = 15\). #### (b) Set Membership and Cardinality **Problem:** Give examples of sets \(A\), \(B\), and \(C\) such that \(A \subseteq B\), \(A \in C\), and \(|A| < |B| < |C|\). **Solution:** - Let \( A = \{1\} \), which has a cardinality of 1. - Let \( B = \{1, 2\} \), which has a cardinality of 2. - Let \( C = \{\{1\}, \{3, 4\}\} \), which has a cardinality of 2. **Justification:** \(A\) is a subset of \(B\) since every element of \(A\) is also in \(B\). \(A\) is an element of \(C\) as a set itself. \(|A| = 1\), \(|B| = 2\), and \(|C| = 2\) satisfy \(|A| < |B| < |C|\). #### (c) Floor Function and Addition **Problem:
Transcribed Image Text:### Examples and Justifications in Set Theory and Real Numbers #### (a) Cartesian Product of Sets **Problem:** Give examples of sets \( X \) and \( Y \) such that \(|X \times Y| = 15 \). **Solution:** Sets \( X \) and \( Y \) must be chosen such that the product of their cardinalities equals 15. For instance: - Let \( X = \{a, b, c\} \), which has a cardinality of 3. - Let \( Y = \{1, 2, 3, 4, 5\} \), which has a cardinality of 5. Thus, \(|X \times Y| = 3 \times 5 = 15\). **Justification:** The Cartesian product of two sets \(X\) and \(Y\) is the set of all ordered pairs \((x, y)\) where \(x \in X\) and \(y \in Y\). Since \(|X| = 3\) and \(|Y| = 5\), the product \(|X \times Y| = 3 \times 5 = 15\). #### (b) Set Membership and Cardinality **Problem:** Give examples of sets \(A\), \(B\), and \(C\) such that \(A \subseteq B\), \(A \in C\), and \(|A| < |B| < |C|\). **Solution:** - Let \( A = \{1\} \), which has a cardinality of 1. - Let \( B = \{1, 2\} \), which has a cardinality of 2. - Let \( C = \{\{1\}, \{3, 4\}\} \), which has a cardinality of 2. **Justification:** \(A\) is a subset of \(B\) since every element of \(A\) is also in \(B\). \(A\) is an element of \(C\) as a set itself. \(|A| = 1\), \(|B| = 2\), and \(|C| = 2\) satisfy \(|A| < |B| < |C|\). #### (c) Floor Function and Addition **Problem:
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