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Recall that the natural numbers are [tex]1, 2, 3, \ldots[/tex], the integers are [tex]\ldots, -3, -2, -1, 0, 1, 2, 3, \ldots[/tex], rational numbers are ratios of integers (with the denominator being non-zero), and real numbers are all numbers corresponding to points on the number line. You can also think of real numbers as repeating or non-repeating decimals. There are more technical definitions which you will learn in real analysis.

Indicate whether the following statements are True (T) or False (F).

1. The sum of two natural numbers is always a natural number.
2. The difference of two natural numbers is always an integer.
3. The quotient of two natural numbers is always a natural number.
4. The quotient of two natural numbers is always a rational number.
5. The difference of two natural numbers is always a natural number.
6. The product of two natural numbers is always a natural number.
7. The ratio of two natural numbers is always positive.



Answer :

GinnyAnswer
Let's go through each statement individually and determine whether it is True (T) or False (F).

1. The sum of two natural numbers is always a natural number.
- Natural numbers are [tex]\(1, 2, 3, \ldots\)[/tex].
- Adding any two natural numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] where [tex]\(a, b \geq 1\)[/tex] will always produce a sum [tex]\(a + b \geq 2\)[/tex], which is a natural number.
- Therefore, this statement is True (T).

2. The difference of two natural numbers is always an integer.
- Natural numbers are [tex]\(\{1, 2, 3, \ldots\}\)[/tex].
- Subtracting any two natural numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] where [tex]\(a - b\)[/tex] might be negative or zero if [tex]\(a < b\)[/tex] or [tex]\(a = b\)[/tex], but the difference will always be an integer.
- Therefore, this statement is True (T).

3. The quotient of two natural numbers is always a natural number.
- Natural numbers are [tex]\(\{1, 2, 3, \ldots\}\)[/tex].
- Dividing any two natural numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] where [tex]\(a/b\)[/tex] may not always result in a natural number, e.g., [tex]\(1/2 = 0.5\)[/tex].
- Therefore, this statement is False (F).

4. The quotient of two natural numbers is always a rational number.
- Rational numbers are ratios of integers where the denominator is not zero.
- Dividing any two natural numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] where [tex]\(a/b\)[/tex] will always produce a rational number, even if it's not an integer.
- Therefore, this statement is True (T).

5. The difference of two natural numbers is always a natural number.
- Natural numbers are [tex]\(\{1, 2, 3, \ldots\}\)[/tex].
- Subtracting any two natural numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] where [tex]\(a - b\)[/tex] might be negative or zero if [tex]\(a \leq b\)[/tex], which are not natural numbers.
- Therefore, this statement is False (F).

6. The product of two natural numbers is always a natural number.
- Natural numbers are [tex]\(\{1, 2, 3, \ldots\}\)[/tex].
- Multiplying any two natural numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] will always produce a product [tex]\(a \times b\)[/tex] which is a natural number.
- Therefore, this statement is True (T).

7. The ratio of two natural numbers is always positive.
- Natural numbers are [tex]\(\{1, 2, 3, \ldots\}\)[/tex].
- Dividing any two natural numbers [tex]\(a\)[/tex] and [tex]\(b\)[/tex] where [tex]\(a, b > 0\)[/tex], their ratio [tex]\(a/b\)[/tex] will always be a positive number.
- Therefore, this statement is True (T).

Considering the analysis above, the results are:
1. True (T)
2. True (T)
3. False (F)
4. True (T)
5. False (F)
6. True (T)
7. True (T)

So the final answers are:
1. True (1)
2. True (1)
3. False (0)
4. True (1)
5. False (0)
6. True (1)
7. True (1)

Which translates to the tuple:
[tex]\((1, 1, 0, 1, 0, 1, 1)\)[/tex]