Let's break this question into two parts, as outlined.
### Part i: Simplify [tex]\( Q \cup Q' \)[/tex]
To address this part, we need to consider the definitions and properties of rational and irrational numbers:
- Rational numbers (denoted by [tex]\( Q \)[/tex]) are numbers that can be expressed as the ratio of two integers [tex]\( \frac{p}{q} \)[/tex] where [tex]\( p \)[/tex] and [tex]\( q \)[/tex] are integers and [tex]\( q \neq 0 \)[/tex].
- Irrational numbers (denoted by [tex]\( Q' \)[/tex]) are numbers that cannot be expressed as a ratio of two integers.
When we take the union of the set of rational numbers [tex]\( Q \)[/tex] with the set of irrational numbers [tex]\( Q' \)[/tex], we are combining these two sets. The union of these two sets encompasses all numbers that are either rational or irrational.
By definition, the set of all real numbers includes both rational and irrational numbers. Therefore:
[tex]\[ Q \cup Q' = \text{All Real Numbers} \][/tex]
### Part ii: Simplify [tex]\( Q \cap Q' \)[/tex]
Next, let's consider the intersection of [tex]\( Q \)[/tex] and [tex]\( Q' \)[/tex]:
- The intersection [tex]\( Q \cap Q' \)[/tex] represents the set of numbers that are both rational and irrational.
However, rational and irrational numbers are mutually exclusive by definition. A number cannot be both rational and irrational simultaneously. Since there is no number that fits both criteria, the intersection of these two sets is empty.
Therefore:
[tex]\[ Q \cap Q' = \text{Empty Set} \][/tex]
### Summary
- For [tex]\( Q \cup Q' \)[/tex], we have determined the set to be all real numbers: [tex]\( Q \cup Q' = \text{All Real Numbers} \)[/tex].
- For [tex]\( Q \cap Q' \)[/tex], we have determined the set to be empty: [tex]\( Q \cap Q' = \text{Empty Set} \)[/tex].
This detailed step-by-step solution simplifies the given mathematical expressions and provides a comprehensive understanding of the properties of rational and irrational numbers.
