Direct Proofs

Familiar Integer Facts We Assume

• the negation of an integer is an integer
• the sum, difference, or product of integers is an integer.

Result 3.4: If $n$ is an odd integer, then $3n+7$ is an even integer.

Proof:

• Assume $n$ is an odd integer.
• So $n = 2k+1, k \in \mathbb{Z}$.
• We want to show that $3n+1 = 2a, where a \in \mathbb{Z}$
• Observe (look!): $$\begin{align} 3n+7 &= 3(2k+1)+7\\ &= 6k+10\\ &= 2(3k+5)\\ \end{align}$$
• Noting that $3k+5 \in \mathbb{Z}$, 3n+7 is even. $\blacksquare$

General Steps

1. Begin with the hypotehsis
2. Deduce what you can from the hypothesis
3. State your goal/conclusion
4. PROVE the goal!
   • Be guided by the upcoming conclusion
5. End with the conclusion.

Result 3.6: If $n$ is an odd integer, then $4n^3+2n-1$ is odd.

Proof:

• Assume $n$ is an odd integer.
• So $n = 2y+1, y \in \mathbb{Z}$.
• We want to show that $4n^3+2n-1 = 2a + 1, a \in \mathbb{Z}$.
• Observe:

$$\begin{align} 4n^3 + 2n - 1 &= 4(2y+1)^3+2(2y+1)-1\\ &= 2\left[2(2y+1)^3+(2y+1) - 1\right] + 1 \end{align}$$

• Thus, $4n^3+2n-1$ is odd. $\blacksquare$