Let's analyze each of the given statements one-by-one to determine their truth values.
1. Statement: 0 is neither a rational number nor an irrational number.
A rational number is defined as any number that can be expressed as the quotient or fraction [tex]\( \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. The number 0 can be expressed as [tex]\( \frac{0}{1} \)[/tex], which fits the definition of a rational number. Therefore, 0 is indeed a rational number.
- Truth value: False
2. Statement: [tex]\( 1 . \overline{3} \)[/tex] is a rational number but not an integer.
The notation [tex]\( 1 . \overline{3} \)[/tex] represents the repeating decimal 1.333..., which is a rational number because it can be expressed as the fraction [tex]\( \frac{4}{3} \)[/tex]. Since 1.333... is not a whole number, it is not an integer.
- Truth value: True
3. Statement: [tex]\( -\sqrt{16} \)[/tex] is an irrational number.
The square root of 16 is 4, so [tex]\( -\sqrt{16} = -4 \)[/tex]. Since -4 is an integer, it is also a rational number. Therefore, [tex]\( -\sqrt{16} \)[/tex] is not an irrational number.
- Truth value: False
4. Statement: [tex]\( \sqrt{2} \)[/tex] is a rational number.
The square root of 2 is known to be an irrational number, as it cannot be expressed as a quotient of two integers.
- Truth value: False
So, the truth values of the statements are:
1. False
2. True
3. False
4. False
Thus, the correct answer is:
[tex]\((0, 1, 0, 0)\)[/tex]
This means only the second statement is true.
