For this homework, you can use the following facts. If it is a definition, you should 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 \( A, B \) be set with \( A \) non-empty.(a) Prove that if \( A \subseteq B \), then \( A, B \) are not disjoint.(b) What if we remove the assumption "A is non-empty"? Is the statement in (a) still correct? Prove or provide a counterexample.

Answered: Image /qna-images/answer/03d51c54-124d-4565-807e-e264caa6a24e.jpg

Let A, B be set with A non-empty. (a) Prove that if A C B, then A, B are not disjoint. (b) What if we remove the assumption "A is non-empty"? Is the state- ment in (a) still correct? Prove or provide a counterexample.

Let A, B be set with A non-empty. (a) Prove that if A C B, then A, B are not disjoint. (b) What if we remove the assumption "A is non-empty"? Is the state- ment in (a) still correct? Prove or provide a counterexample.

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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