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Sagot :
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]
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]
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