To determine which of the statements (a, b, c, d) are true, we need to analyze each statement regarding irrational numbers and their properties.
### Step-by-Step Solution:
1. **Understanding Rational and Irrational Numbers:**
- A **rational number** can be expressed in the form \( \frac{p}{q} \) where \( p \) and \( q \) are integers and \( q \neq 0 \).
- An **irrational number** cannot be expressed in this form. Examples include \( \sqrt{2} \), \( \pi \), and \( e \).
2. **Analyzing Statement A:**
- Statement A claims that "the product of two irrational numbers is always irrational."
- **Example:** Let’s take two irrational numbers, \( \sqrt{2} \) and \( \sqrt{2} \).
- Their product: \( \sqrt{2} \times \sqrt{2} = 2 \), which is a rational number.
- **Conclusion:** Statement A is **false**.
3. **Analyzing Statement B:**
- Statement B claims that "the sum of two irrational numbers is always irrational."
- **Example:** Let’s take \( \sqrt{2} \) and \( -\sqrt{2} \).
- Their sum: \( \sqrt{2} + (-\sqrt{2}) = 0 \), which is a rational number.
- **Conclusion:** Statement B is **false**.
4. **Analyzing Statement C:**
- Statement C claims that "the product of two irrational numbers is always rational."
- **Example:** Let’s take \( \sqrt{2} \) and \( \sqrt{3} \).
- Their product: \( \sqrt{2} \times \sqrt{3} = \sqrt{6} \), which is an irrational number.
- **Conclusion:** Statement C is **false**.
5. **Analyzing Statement D:**
- Since we have not been provided with the content of Statement D, we cannot analyze it directly. However, based on the previous statements, we can conclude that if D states something different from A, B, and C, it may be true.
- If D states that "not all products or sums of irrational numbers yield irrational results," it would be true.
### Final Conclusion:
- Statements A, B, and C are **false**.
- Statement D is likely **true** based on the context provided.
