A1. Study Skills

Reading Proofs

What to do after reading each line: Consider:

• Do I understand the ideas used in that line?
• Do I understand why those ideas have been used?
• How does this link to the other ideas in the proof/other theorems/prior knowledge I have?
• does the self-explanation I have produced help answer those questions?

$\underline{\text{Example:}} \; (\text{Proof by contradiction})$ Statement: \(\forall n \in \mathbb{Z}, 2n+1=a+b+c, \quad \text{where } a, b, c \in \mathbb{Z}, \quad \frac{a}{2}, \frac{b}{2}, \frac{c}{2} \in \mathbb{Z}\)

Assume $x=a+b+c$, let $a=2k, b=2l, c=2m, \quad \text{where } k, l, m \in \mathbb{Z}$ $\therefore 2k + 2l + 2m = 2(k + l + m)$ $\dfrac{x}{2} = k+l+m$, but $k+l+m$ is an integer, $\therefore$ by contradiction, $x$ must be even, thus, no odd integer can be expressed as the sum of three even integers.

Self-explanation technique for understanding proofs

• “The proof aims to show that there exists no integer that can be expressed as a sum of three even integers by assuming it is true and then finding a contradiction.”
• “The three integers can be rewritten as products of 2 and the remaining integer factor.”
• “The formula is then rearranged such that $x$ must also satisfy the definition of an even integer.”
• “Thus, no odd integer can be expressed as the sum of three even integers.”