Homework Help is Here – Start Your Trial Now!
Math
Advanced Math
Prove that if A, B are disjoint and CC B, then A, C are disjoint. You nay assume A, B, C are all non-empty.

Prove that if A, B are disjoint and CC B, then A, C are disjoint. You nay assume A, B, C are all non-empty.

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
See similar textbooks
icon
Related questions
Question
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 numbers. 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 \(U\). (a) We say
Transcribed Image Text: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 numbers. 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 \(U\). (a) We say
**Problem Statement:** Prove that if \( A, B \) are disjoint and \( C \subseteq B \), then \( A, C \) are disjoint. You may assume \( A, B, C \) are all non-empty. **Explanation:** This statement is asking us to establish that under certain conditions, two sets \( A \) and \( C \) have no elements in common. The conditions given are: 1. \( A \) and \( B \) are disjoint, meaning that they have no elements in common. 2. \( C \) is a subset of \( B \), which means every element of \( C \) is also an element of \( B \). Given these conditions, you need to prove that \( A \) and \( C \) are also disjoint, ensuring that \( A \) shares no elements with \( C \). The hint to the proof lies in understanding the relationships between sets \( A \), \( B \), and \( C \). Since \( A \) and \( B \) are disjoint by assumption, it follows logically that any subset of \( B \), including \( C \), will also be disjoint from \( A \). Consider this: - Suppose \( A \) and \( C \) have some common element(s) contrary to what we need to prove. - By the definition of subset, the common elements between \( A \) and \( C \) would also be in \( B \), contradicting the assumption that \( A \) and \( B \) are disjoint. Consequently, the conditions given ensure \( A \) and \( C \) have no elements in common, proving that they are disjoint. This problem is an exercise in understanding the relationships and properties of set operations, particularly subsets and disjoint sets.
Transcribed Image Text:**Problem Statement:** Prove that if \( A, B \) are disjoint and \( C \subseteq B \), then \( A, C \) are disjoint. You may assume \( A, B, C \) are all non-empty. **Explanation:** This statement is asking us to establish that under certain conditions, two sets \( A \) and \( C \) have no elements in common. The conditions given are: 1. \( A \) and \( B \) are disjoint, meaning that they have no elements in common. 2. \( C \) is a subset of \( B \), which means every element of \( C \) is also an element of \( B \). Given these conditions, you need to prove that \( A \) and \( C \) are also disjoint, ensuring that \( A \) shares no elements with \( C \). The hint to the proof lies in understanding the relationships between sets \( A \), \( B \), and \( C \). Since \( A \) and \( B \) are disjoint by assumption, it follows logically that any subset of \( B \), including \( C \), will also be disjoint from \( A \). Consider this: - Suppose \( A \) and \( C \) have some common element(s) contrary to what we need to prove. - By the definition of subset, the common elements between \( A \) and \( C \) would also be in \( B \), contradicting the assumption that \( A \) and \( B \) are disjoint. Consequently, the conditions given ensure \( A \) and \( C \) have no elements in common, proving that they are disjoint. This problem is an exercise in understanding the relationships and properties of set operations, particularly subsets and disjoint sets.
Expert Solution
trending now

Trending now

This is a popular solution!

steps

Step by step

Solved in 2 steps

SEE SOLUTIONCheck out a sample Q&A here
Blurred answer
Recommended textbooks for you
Advanced Engineering Mathematics
Advanced Engineering Mathematics
Advanced Math
ISBN:
9780470458365
Author:
Erwin Kreyszig
Publisher:
Wiley, John & Sons, Incorporated
Numerical Methods for Engineers
Numerical Methods for Engineers
Advanced Math
ISBN:
9780073397924
Author:
Steven C. Chapra Dr., Raymond P. Canale
Publisher:
McGraw-Hill Education
Introductory Mathematics for Engineering Applicat…
Introductory Mathematics for Engineering Applicat…
Advanced Math
ISBN:
9781118141809
Author:
Nathan Klingbeil
Publisher:
WILEY
Mathematics For Machine Technology
Mathematics For Machine Technology
Advanced Math
ISBN:
9781337798310
Author:
Peterson, John.
Publisher:
Cengage Learning,
Basic Technical Mathematics
Basic Technical Mathematics
Advanced Math
ISBN:
9780134437705
Author:
Washington
Publisher:
PEARSON
Topology
Topology
Advanced Math
ISBN:
9780134689517
Author:
Munkres, James R.
Publisher:
Pearson,
SEE MORE TEXTBOOKS