4. Write out the following sets, and give a formal proof that set-builder notation expression (as provided here) is equal to the written-out expression: (1) {x € R : 2² = x} × {x € N: x² = x};
4. Write out the following sets, and give a formal proof that set-builder notation expression (as provided here) is equal to the written-out expression: (1) {x € R : 2² = x} × {x € N: x² = x};
Algebra: Structure And Method, Book 1
(REV)00th Edition
ISBN:9780395977224
Author:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Publisher:Richard G. Brown, Mary P. Dolciani, Robert H. Sorgenfrey, William L. Cole
Chapter2: Working With Real Numbers
Section2.1: Basic Assumptions
Problem 40WE
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Q#4(1)
![**Problem 4: Understanding Set-Builder Notation and Its Equivalence**
In this exercise, the objective is to express the given sets in standard notation and provide a formal proof showing that the set-builder notation provided is equivalent to the conventional written-out expression.
**Expression:**
1. \(\{x \in \mathbb{R} : x^2 = x\} \times \{x \in \mathbb{N} : x^2 = x\}\)
**Explanation:**
- **Set 1: \(\{x \in \mathbb{R} : x^2 = x\}\)**
This set includes all real numbers \(x\) such that \(x^2 = x\). Solving the equation, \(x^2 = x\) can be rewritten as:
\[
x^2 - x = 0 \\
x(x - 1) = 0
\]
Thus, \(x = 0\) or \(x = 1\). Therefore, the set is \(\{0, 1\}\).
- **Set 2: \(\{x \in \mathbb{N} : x^2 = x\}\)**
This set includes all natural numbers \(x\) such that \(x^2 = x\). Again, solving the same equation as before:
\[
x(x - 1) = 0
\]
Thus, for natural numbers, \(x = 1\) (since 0 is not usually included in natural numbers). Therefore, the set is \(\{1\}\).
**Cartesian Product:**
The Cartesian product of the two sets is:
\[
\{0, 1\} \times \{1\}
\]
This results in the ordered pairs \((0, 1)\) and \((1, 1)\).
**Proof:**
To prove the equivalence, we illustrate that the set-builder expressions yield the same pairs when evaluated according to their constraints. Solving \(x^2 = x\) correctly identifies possible values in both real and natural number contexts, thus confirming the drawn conclusion.
In conclusion, the given set-builder notation accurately represents the Cartesian product \(\{(0, 1), (1, 1)\}\).](/v2/_next/image?url=https%3A%2F%2Fcontent.bartleby.com%2Fqna-images%2Fquestion%2F63f643a6-d37e-4098-8dc7-7802480db8dc%2F08f6e436-92ca-4d03-9dab-d159d14a19e5%2Ftl1s8jo_processed.png&w=3840&q=75)
Transcribed Image Text:**Problem 4: Understanding Set-Builder Notation and Its Equivalence**
In this exercise, the objective is to express the given sets in standard notation and provide a formal proof showing that the set-builder notation provided is equivalent to the conventional written-out expression.
**Expression:**
1. \(\{x \in \mathbb{R} : x^2 = x\} \times \{x \in \mathbb{N} : x^2 = x\}\)
**Explanation:**
- **Set 1: \(\{x \in \mathbb{R} : x^2 = x\}\)**
This set includes all real numbers \(x\) such that \(x^2 = x\). Solving the equation, \(x^2 = x\) can be rewritten as:
\[
x^2 - x = 0 \\
x(x - 1) = 0
\]
Thus, \(x = 0\) or \(x = 1\). Therefore, the set is \(\{0, 1\}\).
- **Set 2: \(\{x \in \mathbb{N} : x^2 = x\}\)**
This set includes all natural numbers \(x\) such that \(x^2 = x\). Again, solving the same equation as before:
\[
x(x - 1) = 0
\]
Thus, for natural numbers, \(x = 1\) (since 0 is not usually included in natural numbers). Therefore, the set is \(\{1\}\).
**Cartesian Product:**
The Cartesian product of the two sets is:
\[
\{0, 1\} \times \{1\}
\]
This results in the ordered pairs \((0, 1)\) and \((1, 1)\).
**Proof:**
To prove the equivalence, we illustrate that the set-builder expressions yield the same pairs when evaluated according to their constraints. Solving \(x^2 = x\) correctly identifies possible values in both real and natural number contexts, thus confirming the drawn conclusion.
In conclusion, the given set-builder notation accurately represents the Cartesian product \(\{(0, 1), (1, 1)\}\).
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