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Let's analyze each proposition step by step to determine whether it is true or false.

### Proposition a:
\[
\text{Si } (x - 1)(x + 2) = 0, \text{ entonces } (x - 1) = 0 \text{ y } (x + 2) = 0
\]
This statement is **false**. When two factors are multiplied and the product equals zero, it means **either** \( (x - 1) = 0 \) **or** \( (x + 2) = 0 \). Therefore, the correct conclusion should be:
\[
(x - 1) = 0 \quad \text{or} \quad (x + 2) = 0
\]
Thus, \( x = 1 \) or \( x = -2 \).

### Proposition b:
\[
\text{Todo número } Z \text{ es } Q
\]
This statement is **false**. \( Z \) represents the set of integers, and \( Q \) represents the set of rational numbers. While all integers are rational (since they can be written as a fraction), not all rational numbers are integers. Hence, the reverse is not true: not every rational number is an integer.

### Proposition c:
\[
\text{Si } 2a + b = 0 \text{ y } a - b = 0, \text{ es válido concluir que } 2a + b = a - b
\]
Let's solve this step-by-step:
- From \( a - b = 0 \), we get \( a = b \).
- Substituting \( a = b \) into \( 2a + b = 0 \), we get \( 2b + b = 0 \), which simplifies to \( 3b = 0 \), so \( b = 0 \).
- Since \( b = 0 \), we have \( a = 0 \).

Thus, both sides of the conclusion \( 2a + b = a - b \) become:
\[
0 = 0
\]
Therefore, this statement is **true**.

### Proposition d:
\[
-(3x - y) = y - 3x
\]
Simplifying the left-hand side:
\[
-(3x - y) = -3x + y
\]
The right-hand side is:
\[
y - 3x
\]
Since both sides are equal, the statement is **true**.

### Final answers:
- **a. False**
- **b. False**
- **c. True**
- **d. True**

Would you like any details or further explanations?

Here are 5 related questions to deepen your understanding:
1. What is the difference between the sets \( Z \) and \( Q \)?
2. Can a quadratic equation like \( (x - 1)(x + 2) = 0 \) have two true solutions?
3. How do you solve systems of linear equations like in proposition c?
4. What rules of distributivity and factorization are used to simplify expressions like in proposition d?
5. How can one check if a mathematical proposition is logically valid?

**Tip**: When solving equations, it's important to distinguish between "and" and "or" conditions to avoid incorrect conclusions.

En cada caso, determine si la proposición es verdadera o falsa.
a. Si (x-1)(x+2)=0, entonces (x-1)=0 y (x+2)=0.
b. Todo número Z es Q.
c. Si 2a + b = 0 y a - b = 0, es válido concluir que: 2a + b = a - b.
d. -(3x-y) = y - 3x.

Learn how to determine the truth value of algebraic and set theory propositions. This includes solving quadratic equations, understanding sets Z and Q, and simplifying linear equations. Suitable for Grades 8-10.

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