We are asked to prove the following statement:

For all $x, y \in \mathbb{R}$, if $xy = 0$, then either $x = 0$ or $y = 0$.

Step-by-Step Solution:

Step 1: Restate the Problem

We need to show that if the product of two real numbers is zero, then at least one of the numbers must be zero.

That is: $xy = 0 \implies x = 0 \text{ or } y = 0.$

This is a basic property of real numbers, commonly referred to as the zero product property.

Step 2: Proof by Contrapositive (Optional Method)

One approach is to prove this by contradiction. Assume the opposite of the conclusion, i.e., both $x \neq 0$ and $y \neq 0$, and show that this leads to a contradiction.

Step 2.1: Assume $x \neq 0$ and $y \neq 0$

If $x \neq 0$ and $y \neq 0$, then both $x$ and $y$ are non-zero real numbers.

Step 2.2: Multiply $x$ and $y$

Since $x \neq 0$ and $y \neq 0$, multiplying them will give: $xy \neq 0.$ This contradicts the assumption $xy = 0$.

Thus, the assumption that both $x$ and $y$ are non-zero must be false. Therefore, it must be the case that either $x = 0$ or $y = 0$.

Step 3: Direct Proof (Simple Method)

Alternatively, we can approach this directly:

Step 3.1: Case 1: $x = 0$

If $x = 0$, then clearly $xy = 0 \cdot y = 0$, which satisfies the condition $xy = 0$.

Step 3.2: Case 2: $y = 0$

If $y = 0$, then clearly $xy = x \cdot 0 = 0$, which also satisfies the condition $xy = 0$.

Step 3.3: Case 3: Neither $x = 0$ nor $y = 0$

If both $x \neq 0$ and $y \neq 0$, as shown earlier, $xy \neq 0$, contradicting the assumption that $xy = 0$.

Conclusion:

We have shown that if $xy = 0$, then either $x = 0$ or $y = 0$. This proves the original statement.

Would you like more details or have any specific questions about this proof?

Related Questions:

1. What is the significance of the zero product property in algebra?
2. How does this proof apply to complex numbers or other number systems?
3. Can we extend this property to polynomials?
4. What are the conditions under which the zero product property might fail (e.g., in different algebraic structures)?
5. How does this concept relate to solving quadratic or higher-degree equations?

Tip:

Whenever you encounter a product of numbers equaling zero, always consider the possibility that one or more of the factors must be zero. This is a powerful tool in both algebra and calculus.