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No calculators may be used unless otherwise stated. 1. Give an example to show that: a) \( \mathbb{N} \) is not closed under subtraction b) Not every Whole Number is a Natural Num c) \( \mathbb{Z} \) is not closed under division d) \( Q \) is not closed under division.

Ask by Parry Garza. in South Africa
Jan 20,2025

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Answer

- **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.

Solution

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.

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To illustrate that \( \mathbb{N} \) is not closed under subtraction, take two natural numbers: \( 3 \) and \( 5 \). When you subtract \( 5 - 3 \), you get \( -2 \), which is not a natural number. So, \( \mathbb{N} \) certainly isn’t closed under subtraction! For the second point, consider the number \( 0 \). While every natural number is a positive integer starting from \( 1 \) onward, \( 0 \) is classified as a whole number but does not fall under the category of natural numbers. This distinction shows that not every whole number is a natural number. Now, let's say you take two integers: \( 4 \) and \( 0 \) from \( \mathbb{Z} \). If you try to divide \( 4 \) by \( 0 \), the operation produces an undefined result. Thus, \( \mathbb{Z} \) is not closed under division because we cannot divide by zero! Lastly, consider the rational numbers \( \frac{1}{2} \) and \( 0 \). If you try dividing \( \frac{1}{2} \) by \( 0 \), it leads to an undefined result again. This indicates that \( \mathbb{Q} \) is not closed under division, as there are valid rational numbers that cannot be divided by zero.

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