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Chapter 1: Problem 1

Let \(P\) be a subset of the real numbers \(R\) such that (i) \(1 \in P\) (ii) if \(a, b \in P\) then \(a+b \in P\), and (iii) for each \(x \in R\), one and only one of the three statements, \(x \in P, x=0\), or \(-x \in P\) is true. Define \(Q=\\{(a, b) \mid(a, b) \in R \times R\) and \(a-b \in P\\} .\) Prove that \(Q\) is a transitive relation.

Short Answer

Expert verified
Q is transitive because if \((a,b) \in Q\) and \((b,c) \in Q\), then \(a-c \in P\), implying \((a,c) \in Q\).

Step by step solution

01

Understand the Definitions

Start by understanding the properties of the set \(P\) and the relation \(Q\). \(P\) contains 1 and is closed under addition. Additionally, exactly one of \(x \in P\), \(x = 0\), or \(-x \in P\) is true for any real number \(x\). The relation \(Q\) consists of pairs \((a, b)\) for which \(a-b \in P\). Our goal is to show that \(Q\) is transitive.
02

Recall the Definition of a Transitive Relation

A relation \(Q\) is transitive if for all \((a, b), (b, c) \in Q\), it follows that \((a, c) \in Q\). This means that if two elements are related through a common element, then the first and the third elements should also be directly related.
03

Assume the Hypothesis

Assume \((a, b) \in Q\) and \((b, c) \in Q\). This implies that \(a - b \in P\) and \(b - c \in P\) by the definition of \(Q\). We must prove that \((a, c) \in Q\), which means \(a - c \in P\).
04

Use Closure Property of P

Since \(a - b \in P\) and \(b - c \in P\), their sum \((a - b) + (b - c) = a - c\) is also in \(P\) due to the closure property: if \(x, y \in P\) then \(x + y \in P\).
05

Conclude Transitivity of Q

Since \(a - c \in P\) by the closure property, it follows that \((a, c) \in Q\). Therefore, if \((a, b) \in Q\) and \((b, c) \in Q\), then \((a, c) \in Q\). This confirms that \(Q\) is a transitive relation.

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Closure Property
In topology and set theory, the closure property plays a vital role. It is a way of saying that a set is equipped to "handle" its operations internally.
For any real numbers you choose within a set that satisfies closure for an operation, the result of that operation will also lie within the set.
In the context of our exercise, if set \( P \) includes two elements \( a \) and \( b \), then their sum \( a + b \) must also be in \( P \).
This property ensures that when you combine elements of the set \( P \) through addition, you will be able to stay in the realm of \( P \).
  • For example, if \( 1 \) and \( 2 \) are in \( P \), then \( 1 + 2 = 3 \) must also be in \( P \).
  • The closure property helps maintain the robustness of set operations, particularly demonstrating their internal consistency.
Subset of Real Numbers
A subset of real numbers is a part of the entire collection of real number values.
In mathematics, any set that includes real numbers can be a considered subset if it contains some, but not necessarily all, of the elements from the real numbers set.
Specifically, in our exercise, the subset \( P \) is defined under two unique conditions:
  • \( 1 \) is an element of \( P \).
  • If any two elements \( a \) and \( b \) are in \( P \), then their sum \( a + b \) must also be in \( P \).
Therefore, \( P \) showcases a specific organized structure of real number values that follow the given rules. This helps mathematicians understand and explore the properties, behavior, and operations on these numbers.
Properties of Set P
Discussing the properties of set \( P \) allows for deeper insight into its structural behavior. In the given exercise, set \( P \) holds some distinct characteristics.
Let's break these down:
  • Contains 1: \( 1 \) is always included, which is the foundational characteristic building the rest of the properties.
  • Closed under Addition: This property means that if any two numbers are present in \( P \), their sum should not leave \( P \), emphasizing orderly consistency and rule-abiding nature in arithmetic operations.
  • Mutual Exclusivity Property: For any real number \( x \), only one of three conditions will hold: \( x \) is in \( P \), \( x = 0 \), or \(-x\) is in \( P \). This ensures a clear structure and organization in defining which numbers can appear in the set \( P \).
Understanding these features is key to exploring how the set behaves with arithmetic operations, especially when addressing relations like \( Q \).
Relation Definition
A relation in mathematics generally describes how two sets are connected and behave in regard to each other.
In our specific case, the relation \( Q \) is defined as a pairing of real numbers based on the properties of the set \( P \).
To elaborate, \( Q \) consists of pairs \( (a, b) \) where the difference \( a - b \) is contained in \( P \).
This relational pattern helped us create a topology where transitivity, a renowned characteristic, could be further examined.
  • Transitive Relation: A relation is transitive if whenever a first element is related to a second one and the second is related to a third, the first is also related to the third (using the same relation).
  • For \( Q \), this specifically means that if both \( a - b \) and \( b - c \) fall within \( P \), then \( a - c \) should also land in \( P \) to maintain the relation's transitivity.
  • This illustrates a sense of "path continuity" within the set \( P \) using the operation of subtraction among its elements, but maintaining the feeling of unity and structured coherence.

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Most popular questions from this chapter

Let \(f: X \rightarrow Y\) be a function from a set \(X\) onto a set \(Y\). Let \(R\) be the subset of \(X \times X\) consisting of those pairs \(\left(x, x^{\prime}\right)\) such that \(f(x)=f\left(x^{\prime}\right)\). Prove that \(R\) is an equivalence relation. Let \(\pi: X \rightarrow X / R\) be the projection. Verify that, if \(\alpha \in X / R\) is an equivalence class, to define \(F(\alpha)=f(a)\), whenever \(\alpha=\pi(a)\), establishes a well-defined function \(F: X / R \rightarrow Y\) which is oneone and onto.

Let \(f: A \rightarrow B\) be given and let \(\left\\{Y_{\alpha}\right\\}_{a \in I}\) be an indexed family of subsets of \(B\). Prove: (a) \(f^{-1}\left(\bigcup_{a \in I} Y_{\alpha}\right)=\bigcup_{a \in I} f^{-1}\left(Y_{\alpha}\right)\). (b) \(f^{-1}\left(\cap_{\propto \in I} Y_{\alpha}\right)=\bigcap_{\alpha \in I} f^{-1}\left(Y_{\alpha}\right)\). (c) If \(X\) is a subset of \(B\) then \(f^{-1}(C(X))=C\left(f^{-1}(X)\right)\). (d) If \(X\) is a subset of \(A\), and \(Y\) is a subset of \(B\), then \(f\left(X \cap f^{-1}(Y)\right)=f(X) \cap Y\).

Let \(A \subset S, B \subset S\). Prove the following: (a) \(A \subset B\) if and only if \(A \cup B=B\). (b) \(A \subset B\) if and only if \(A \cap B=A\). (c) \(A \subset C(B)\) if and only if \(A \cap B=\emptyset\). (d) \(C(A) \subset B\) if and only if \(A \cup B=S\). (e) \(A \subset B\) if and only if \(C(B) \subset C(A)\). (f) \(A \subset C(B)\) if and only if \(B \subset C(A)\).

Let \(X\) be the set of functions from the real numbers into the real numbers possessing continuous derivatives. Let \(R\) be the subset of \(X \times\) \(X\) consisting of those pairs \((f, g)\) such that \(D f=D g\) where \(D\) maps a function into its derivative. Prove that \(R\) is an equivalence relation and describe an equivalence set \(\pi(f)\).

Let \(f: A \rightarrow B\) be given and let \(\left\\{X_{\alpha}\right\\}_{\alpha \in I}\) be an indexed family of subsets of \(A\). Prove: (a) \(f\left(\bigcup_{a \in I} X_{\alpha}\right)=\bigcup_{a \in I} f\left(X_{\alpha}\right)\). (b) \(f\left(\cap_{\alpha \in I} X_{\alpha}\right) \subset \cap_{\alpha \in I} f\left(X_{\alpha}\right)\). (c) If \(f: A \rightarrow B\) is one-one, then \(f\left(\cap_{\alpha \in I} X_{\alpha}\right)=\bigcap_{\alpha \in I} f\left(X_{\alpha}\right)\).
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