No calculators may be used unless otherwise stated.

Give an example to show that:
a) is not closed under subtraction
b) Not every Whole Number is a Natural Num
c) is not closed under division
d) is not closed under division.

Ask by Parry Garza. in South Africa

Jan 20,2025

Question

No calculators may be used unless otherwise stated.

Give an example to show that:
a) is not closed under subtraction
b) Not every Whole Number is a Natural Num
c) is not closed under division
d) is not closed under division.

Ask by Parry Garza. in South Africa

Jan 20,2025

UpStudy AI · Accepted Answer

Sure, let's go through each part one by one with appropriate examples:

### 1. a) $$ \mathbb{N} $$ is Not Closed Under Subtraction

**Explanation:**  
The set of natural numbers ($$ \mathbb{N} $$) typically includes positive integers starting from 1 (i.e., $$ \mathbb{N} = \{1, 2, 3, \dots\} $$). For a set to be closed under an operation, performing that operation on any two elements of the set should result in another element that also belongs to the set.

**Example:**  
Consider the natural numbers 2 and 3.

$$ 2 - 3 = -1 $$

**Conclusion:**  
$$-1$$ is not a natural number since $$ \mathbb{N} $$ contains only positive integers. Therefore, $$ \mathbb{N} $$ is not closed under subtraction.

---

### 1. b) Not Every Whole Number is a Natural Number

**Explanation:**  
Whole numbers generally include all natural numbers along with zero (i.e., Whole Numbers = $$ \{0, 1, 2, 3, \dots\} $$). Natural numbers typically start from 1.

**Example:**  
Consider the number 0.

**Conclusion:**  
0 is a whole number but **not** a natural number. This demonstrates that not every whole number is a natural number.

---

### 1. c) $$ \mathbb{Z} $$ is Not Closed Under Division

**Explanation:**  
The set of integers ($$ \mathbb{Z} $$) includes all positive and negative whole numbers as well as zero (i.e., $$ \mathbb{Z} = \{\dots, -2, -1, 0, 1, 2, \dots\} $$). For a set to be closed under division, dividing any two integers (except by zero) should result in another integer.

**Example:**  
Consider the integers 1 and 2.

$$ \frac{1}{2} = 0.5 $$

**Conclusion:**  
0.5 is not an integer. Therefore, $$ \mathbb{Z} $$ is not closed under division.

---

### 1. d) $$ \mathbb{Q} $$ is Not Closed Under Division

**Explanation:**  
The set of rational numbers ($$ \mathbb{Q} $$) includes all numbers that can be expressed as the quotient of two integers, where the denominator is not zero (i.e., $$ \mathbb{Q} = \left\{ \frac{a}{b} \mid a, b \in \mathbb{Z}, b \neq 0 \right\} $$). While $$ \mathbb{Q} $$ is generally closed under division, division by zero is undefined.

**Example:**  
Consider the rational numbers 1 and 0.

$$ \frac{1}{0} $$

**Conclusion:**  
Division by zero is undefined and does not result in a rational number. Therefore, $$ \mathbb{Q} $$ is **not** closed under division when considering all possible cases, including division by zero.

---

**Summary:**  
- **a)** Subtracting 3 from 2 in $$ \mathbb{N} $$ results in -1, which is not in $$ \mathbb{N} $$.
- **b)** 0 is a whole number but not a natural number.
- **c)** Dividing 1 by 2 in $$ \mathbb{Z} $$ results in 0.5, which is not an integer.
- **d)** Dividing 1 by 0 is undefined, showing $$ \mathbb{Q} $$ is not closed under division in this case.
- **a)** Subtracting 3 from 2 in natural numbers gives -1, which isn't a natural number. - **b)** 0 is a whole number but not a natural number. - **c)** Dividing 1 by 2 in integers results in 0.5, which isn't an integer. - **d)** Dividing 1 by 0 is undefined, showing rational numbers aren't closed under division in this case.

No calculators may be used unless otherwise stated.

Give an example to show that:
a) is not closed under subtraction
b) Not every Whole Number is a Natural Num
c) is not closed under division
d) is not closed under division.

Upstudy AI Solution

Answer

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No calculators may be used unless otherwise stated. Give an example to show that: a) is not closed under subtraction b) Not every Whole Number is a Natural Num c) is not closed under division d) is not closed under division.

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No calculators may be used unless otherwise stated.

Give an example to show that:
a) is not closed under subtraction
b) Not every Whole Number is a Natural Num
c) is not closed under division
d) is not closed under division.