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Let A, B, C be non-empty subsets of the universal set U such that (An B) CC. Prove that AC BUC.

Let A, B, C be non-empty subsets of the universal set U such that (An B) CC. Prove that AC BUC.

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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### Math 135: HW 0 For this homework, use the following facts. If it is a definition, mention “by definition” or “by definition of XXX (e.g. even)”. If it is not a definition, you do NOT have to cite. 1. **Basic algebra**, such as \(2 + 2 = 4\), \(1 - 3 = -2\), \(2 \cdot 4 = 8\). This includes subtracting an integer, or dividing by a non-zero rational/real/complex number. 2. **Common knowledge of whether numbers are integer/rational**, e.g., \(\frac{1}{2}\) is not an integer, \(\pi\) is not rational, all integers are rational, all rational numbers are real, all real numbers are complex, etc. 3. **Associative law of addition**: \((a+b) + c = a + (b+c)\), and **associative law of multiplication**: \((ab)c = a(bc)\), for all \(a, b, c \in \mathbb{C}\). 4. **Commutative law of addition**: \(a + b = b + a\), and **commutative law of multiplication**: \(ab = ba\), for all \(a, b \in \mathbb{C}\). 5. **Distributive law**: For all \(a, b, c \in \mathbb{C}\), we have \((a+b)(c+d) = ac + ad + bc + bd\). In particular, \((a+b)^2 = a^2 + 2ab + b^2\). 6. An integer \(n\) is **even** if there exists an integer \(k\) such that \(n = 2k\). 7. An integer \(n\) is **odd** if there exists an integer \(k\) such that \(n = 2k + 1\). 8. All integers are either even or odd. 9. A real number \(x\) is **positive** if \(x > 0\), and **negative** if \(x < 0\). 10. All real numbers are either positive, negative, or 0. 11. Let \(A, B\) be subsets of a universal set \(U\).
Transcribed Image Text:### Math 135: HW 0 For this homework, use the following facts. If it is a definition, mention “by definition” or “by definition of XXX (e.g. even)”. If it is not a definition, you do NOT have to cite. 1. **Basic algebra**, such as \(2 + 2 = 4\), \(1 - 3 = -2\), \(2 \cdot 4 = 8\). This includes subtracting an integer, or dividing by a non-zero rational/real/complex number. 2. **Common knowledge of whether numbers are integer/rational**, e.g., \(\frac{1}{2}\) is not an integer, \(\pi\) is not rational, all integers are rational, all rational numbers are real, all real numbers are complex, etc. 3. **Associative law of addition**: \((a+b) + c = a + (b+c)\), and **associative law of multiplication**: \((ab)c = a(bc)\), for all \(a, b, c \in \mathbb{C}\). 4. **Commutative law of addition**: \(a + b = b + a\), and **commutative law of multiplication**: \(ab = ba\), for all \(a, b \in \mathbb{C}\). 5. **Distributive law**: For all \(a, b, c \in \mathbb{C}\), we have \((a+b)(c+d) = ac + ad + bc + bd\). In particular, \((a+b)^2 = a^2 + 2ab + b^2\). 6. An integer \(n\) is **even** if there exists an integer \(k\) such that \(n = 2k\). 7. An integer \(n\) is **odd** if there exists an integer \(k\) such that \(n = 2k + 1\). 8. All integers are either even or odd. 9. A real number \(x\) is **positive** if \(x > 0\), and **negative** if \(x < 0\). 10. All real numbers are either positive, negative, or 0. 11. Let \(A, B\) be subsets of a universal set \(U\).
Let \( A, B, C \) be non-empty subsets of the universal set \( U \) such that \( (A \cap B)^c \subseteq C \). Prove that \( A \subseteq B \cup C \).
Transcribed Image Text:Let \( A, B, C \) be non-empty subsets of the universal set \( U \) such that \( (A \cap B)^c \subseteq C \). Prove that \( A \subseteq B \cup C \).
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