To prove that if $x$ is an even integer, then $x - 3$ is an odd integer, we will use the definitions of even and odd integers and basic properties of arithmetic. Here's the formal proof:

Definitions

1. An integer $x$ is even if there exists an integer $k$ such that $x = 2k$.
2. An integer $y$ is odd if there exists an integer $m$ such that $y = 2m + 1$.

Proof

1. Assume $x$ is an even integer.
2. By the definition of an even integer, there exists an integer $k$ such that: $x = 2k$
3. Consider $x - 3$: $x - 3 = 2k - 3$
4. We need to show that $2k - 3$ is an odd integer.
5. To show $2k - 3$ is odd, we need to express it in the form $2m + 1$ for some integer $m$.

Express $2k - 3$ in the Form $2m + 1$

1. Rewrite $2k - 3$ as follows: $2k - 3 = 2(k - 2) + 1$
2. Let $m = k - 2$. Notice that since $k$ is an integer, $k - 2$ is also an integer (since the set of integers is closed under subtraction). Thus, $m$ is an integer.
3. Therefore: $2k - 3 = 2m + 1$ where $m = k - 2$.

Conclusion

Since $x - 3$ can be expressed in the form $2m + 1$, where $m$ is an integer, $x - 3$ is an odd integer.

Final Statement

Therefore, if $x$ is an even integer, then $x - 3$ is an odd integer.

Would you like more details or have any questions?

Here are 5 questions you might want to ask next:

1. Can you provide an example of an even integer and show the calculation for $x - 3$ being odd?
2. How can we prove that the sum of two odd integers is even?
3. Can you explain the difference between a direct proof and a proof by contradiction?
4. How can we formally prove that the product of two even integers is even?
5. What is the significance of proving statements about even and odd integers in mathematics?

Tip: When proving statements about numbers, always start with the definitions and use algebraic manipulations to show the required property.