1. Suppose we would like to define a function f : Q→ Z by the formula f(r) = a +b, where a, b € Z are integers with b 0 such that r = a/b (every rational number can be represented as such a fraction, by definition). This is an instance of a formula which does not yield a well-defined function. What do you think is meant by this phrase? What property in the definition of a function from Q to Z does the displayed formula fail to satisfy? Justify your answer with an explicit numerical example.
1. Suppose we would like to define a function f : Q→ Z by the formula f(r) = a +b, where a, b € Z are integers with b 0 such that r = a/b (every rational number can be represented as such a fraction, by definition). This is an instance of a formula which does not yield a well-defined function. What do you think is meant by this phrase? What property in the definition of a function from Q to Z does the displayed formula fail to satisfy? Justify your answer with an explicit numerical example.
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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![### Cartesian Product and Functions
**Cartesian Product Definition:**
If \( A \) and \( B \) are sets, the **cartesian product** of \( A \) and \( B \), denoted \( A \times B \) (“A times B”), is the set consisting of all ordered pairs \((a, b)\) with \( a \in A \) and \( b \in B \). The elements \((a_1, b_1)\) and \((a_2, b_2)\) of \( A \times B \) are equal if and only if \( a_1 = a_2 \) and \( b_1 = b_2 \). Note that \( A \times B \) is the empty set if either \( A = \emptyset \) or \( B = \emptyset \); the symbol \(\emptyset\) always denotes the empty set, and should not be used for the number 0).
**Function Definition:**
A **function** (or map, or mapping) from \( A \) to \( B \) is a subset of \( A \times B \) with the property that, for every \( a \in A \), there is exactly one \( b \in B \) such that \((a, b) \in f\). It is customary to denote this \( b \) by \( f(a) \), and to refer to \( f(a) \) as the image of \( a \) under \( f \), or the value of \( a \) at \( f \). To indicate that \( f \) is a function from \( A \) to \( B \), we write \( f : A \rightarrow B \); the set \( A \) is the **domain** of \( f \), and \( B \) is the **codomain** of \( f \); the range of \( f \) is the set \( f(A) \) defined by:
\[ f(A) = \{ f(a) : a \in A \}. \]
**Equality of Functions:**
Two functions are equal if and only if they have the same domain, the same codomain, and the same value at each element of their common domain. This means that, for instance, if we define functions \( f : \mathbb{R} \to \mathbb{R} \) and](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2Febb913e1-4986-4d74-b6ce-ad576ddf43d3%2F15faf554-f987-4a8b-a406-18edaaa7aaa8%2Fjpyz2c_processed.png&w=3840&q=75)
Transcribed Image Text:### Cartesian Product and Functions
**Cartesian Product Definition:**
If \( A \) and \( B \) are sets, the **cartesian product** of \( A \) and \( B \), denoted \( A \times B \) (“A times B”), is the set consisting of all ordered pairs \((a, b)\) with \( a \in A \) and \( b \in B \). The elements \((a_1, b_1)\) and \((a_2, b_2)\) of \( A \times B \) are equal if and only if \( a_1 = a_2 \) and \( b_1 = b_2 \). Note that \( A \times B \) is the empty set if either \( A = \emptyset \) or \( B = \emptyset \); the symbol \(\emptyset\) always denotes the empty set, and should not be used for the number 0).
**Function Definition:**
A **function** (or map, or mapping) from \( A \) to \( B \) is a subset of \( A \times B \) with the property that, for every \( a \in A \), there is exactly one \( b \in B \) such that \((a, b) \in f\). It is customary to denote this \( b \) by \( f(a) \), and to refer to \( f(a) \) as the image of \( a \) under \( f \), or the value of \( a \) at \( f \). To indicate that \( f \) is a function from \( A \) to \( B \), we write \( f : A \rightarrow B \); the set \( A \) is the **domain** of \( f \), and \( B \) is the **codomain** of \( f \); the range of \( f \) is the set \( f(A) \) defined by:
\[ f(A) = \{ f(a) : a \in A \}. \]
**Equality of Functions:**
Two functions are equal if and only if they have the same domain, the same codomain, and the same value at each element of their common domain. This means that, for instance, if we define functions \( f : \mathbb{R} \to \mathbb{R} \) and
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