To prove that the expression $(\epsilon + 1)^3 - 5\epsilon^2 + 6$ is odd for all $\epsilon \in \mathbb{N}$ (the set of natural numbers), we will use formal mathematical deduction.

Step-by-Step Proof

1. Expression Analysis:
   Consider the expression: $f(\epsilon) = (\epsilon + 1)^3 - 5\epsilon^2 + 6.$

2. Expand the Cubic Term:
   Start by expanding $(\epsilon + 1)^3$: $(\epsilon + 1)^3 = \epsilon^3 + 3\epsilon^2 + 3\epsilon + 1.$

3. Substitute and Simplify:
   Substitute the expanded term into the expression: $f(\epsilon) = (\epsilon^3 + 3\epsilon^2 + 3\epsilon + 1) - 5\epsilon^2 + 6.$ Simplify the expression by combining like terms: $f(\epsilon) = \epsilon^3 - 2\epsilon^2 + 3\epsilon + 7.$

4. Determine the Parity of $f(\epsilon)$:
   To determine if $f(\epsilon)$ is odd, we need to show that $f(\epsilon) \mod 2 = 1$.

   Evaluate each term modulo 2:

   • $\epsilon^3 \mod 2 = \epsilon \mod 2$ (since the cube of an integer has the same parity as the integer itself).
   • $2\epsilon^2 \mod 2 = 0$ (since $2\epsilon^2$ is always even).
   • $3\epsilon \mod 2 = (1\epsilon) \mod 2 = \epsilon \mod 2$ (since $3 \equiv 1 \pmod{2}$).
   • $7 \mod 2 = 1$ (since 7 is odd).

   Substitute these into the expression: $f(\epsilon) \equiv (\epsilon - 0 + \epsilon + 1) \mod 2.$ Simplify: $f(\epsilon) \equiv (2\epsilon + 1) \mod 2.$

5. Conclude the Parity:
   Since $2\epsilon \mod 2 = 0$, we have: $f(\epsilon) \equiv 0 + 1 \equiv 1 \mod 2.$

   Therefore, $f(\epsilon)$ is odd for all $\epsilon \in \mathbb{N}$.

Conclusion

We have shown that $(\epsilon + 1)^3 - 5\epsilon^2 + 6$ is odd for all $\epsilon \in \mathbb{N}$ using formal logical deduction and properties of congruences.

Would you like more details on any part of the proof, or do you have any other questions?

Here are 5 related questions that might interest you:

1. How do we prove the parity of a polynomial function in general?
2. What is the significance of modular arithmetic in number theory?
3. How can we extend this method to prove properties of more complex expressions?
4. Can this technique be used to prove properties for expressions over other domains (e.g., integers)?
5. What is the relevance of proving properties for all natural numbers in mathematical logic?

Tip: When proving properties involving parity (odd or even), modular arithmetic is a powerful tool to simplify calculations!