To determine which sets include the number [tex]\( \frac{5}{9} \)[/tex], we need to understand the properties of each of these sets of numbers:
1. Natural Numbers: Natural numbers are the set of all positive integers starting from 1 (1, 2, 3, ...). Since [tex]\( \frac{5}{9} \)[/tex] is a fraction and not a positive integer, it is not a natural number.
- Set membership: [tex]\( \frac{5}{9} \notin \mathbb{N} \)[/tex]
2. Whole Numbers: Whole numbers are the set of non-negative integers, starting from 0 (0, 1, 2, ...,). Again, since [tex]\( \frac{5}{9} \)[/tex] is not an integer but a fraction, it is not a whole number.
- Set membership: [tex]\( \frac{5}{9} \notin \mathbb{W} \)[/tex]
3. Integers: Integers include all positive and negative whole numbers including zero (..., -2, -1, 0, 1, 2, ...). Since [tex]\( \frac{5}{9} \)[/tex] is not a whole number, it is not an integer either.
- Set membership: [tex]\( \frac{5}{9} \notin \mathbb{Z} \)[/tex]
4. Rational Numbers: Rational numbers are those numbers that can be expressed as the quotient or fraction [tex]\( \frac{p}{q} \)[/tex] of two integers, where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex]. Since [tex]\( \frac{5}{9} \)[/tex] can be expressed in this form (where 5 and 9 are both integers and 9 is not zero), it is a rational number.
- Set membership: [tex]\( \frac{5}{9} \in \mathbb{Q} \)[/tex]
5. Irrational Numbers: Irrational numbers are those numbers that cannot be expressed as a simple fraction of two integers. Since [tex]\( \frac{5}{9} \)[/tex] can be expressed as a fraction, it is not an irrational number.
- Set membership: [tex]\( \frac{5}{9} \notin \mathbb{I} \)[/tex]
6. Real Numbers: Real numbers include all rational and irrational numbers. Since [tex]\( \frac{5}{9} \)[/tex] is a rational number, it is also a real number.
- Set membership: [tex]\( \frac{5}{9} \in \mathbb{R} \)[/tex]
In summary, [tex]\( \frac{5}{9} \)[/tex] falls under the sets of rational numbers and real numbers, but not under the sets of natural numbers, whole numbers, integers, or irrational numbers.
So, the result is:
- Natural numbers: False
- Whole numbers: False
- Integers: False
- Rational numbers: True
- Irrational numbers: False
- Real numbers: True
