To prove the contrapositive of the statement "If $p^2 + q^2 + 1$ is odd, then $p + q$ is even," we need to restate the contrapositive.

The Contrapositive:

The contrapositive of a statement of the form "If A, then B" is "If not B, then not A."

Given the original statement: Original statement: If $p^2 + q^2 + 1$ is odd, then $p + q$ is even.

The contrapositive would be: Contrapositive: If $p + q$ is odd, then $p^2 + q^2 + 1$ is even.

Proof of the Contrapositive:

We want to prove that if $p + q$ is odd, then $p^2 + q^2 + 1$ is even.

Step 1: Express $p + q$ being odd.

• If $p + q$ is odd, then one of $p$ or $q$ must be odd, and the other must be even. This is because the sum of an odd and even number is odd.

Thus, we consider the two possible cases:

• Case 1: $p$ is odd and $q$ is even.
• Case 2: $p$ is even and $q$ is odd.

Step 2: Evaluate $p^2 + q^2 + 1$ in both cases.

• Case 1: $p$ is odd and $q$ is even.

  • If $p$ is odd, then $p^2$ is odd (since the square of an odd number is odd).
  • If $q$ is even, then $q^2$ is even (since the square of an even number is even).
  • Thus, $p^2 + q^2$ is odd + even = odd.
  • Therefore, $p^2 + q^2 + 1$ is odd + 1 = even.

• Case 2: $p$ is even and $q$ is odd.

  • If $p$ is even, then $p^2$ is even (since the square of an even number is even).
  • If $q$ is odd, then $q^2$ is odd (since the square of an odd number is odd).
  • Thus, $p^2 + q^2$ is even + odd = odd.
  • Therefore, $p^2 + q^2 + 1$ is odd + 1 = even.

Step 3: Conclusion.

In both cases, when $p + q$ is odd, $p^2 + q^2 + 1$ is even. Hence, the contrapositive of the original statement is true.

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Follow-up Questions:

1. What is the difference between a statement and its contrapositive?
2. How would you prove the original statement directly, without using the contrapositive?
3. Can you think of a counterexample where the contrapositive fails for a different type of problem?
4. How do you prove the contrapositive for statements involving inequalities or other operations?
5. What are some common mistakes when proving or applying contrapositives?

Tip:

When dealing with mathematical proofs, always check whether working with the contrapositive might simplify the process. It is often easier to prove the contrapositive than the original statement directly, especially in logical or modular arithmetic problems.