# 反證法

(由矛盾證明跳轉過嚟)

## 真值表

${\displaystyle p}$  ${\displaystyle q}$  ${\displaystyle p}$ ${\displaystyle q}$
T T T
T F F
F T T
F F T

## 簡單例子

{\displaystyle {\begin{aligned}n^{2}&=(2k+1)^{2}\\&=4k^{2}+2k+1\\\end{aligned}}}

## ${\displaystyle {\sqrt {2}}}$ 係非有理數

{\displaystyle {\begin{aligned}{\sqrt {2}}&={\frac {m}{n}}\\({\sqrt {2}})^{2}&={\bigl (}{\frac {m}{n}}{\bigr )}^{2}\\2&={\frac {m^{2}}{n^{2}}}\\2n^{2}&=m^{2}\end{aligned}}}

{\displaystyle {\begin{aligned}2n^{2}&=m^{2}\\2n^{2}&=(2k)^{2}\\2n^{2}&=4k^{2}\\n^{2}&=2k^{2}\end{aligned}}}

${\displaystyle {\sqrt {2}}={\frac {m}{n}}={\frac {2k}{2j}}={\frac {k}{j}}}$

## 反證法嘅步驟

1. 寫低要證明嘅命題。
2. 再寫低頭先嗰句命題嘅相反，再假設呢句新命題係啱。
3. 利用呢條假設，推斷啲野出嚟。
4. 搵下有咩矛盾。
5. 如果個假設引致矛盾，咁就可以話個假設係錯。咁嘅話個假設嘅相反－即係想證明嗰句命題－就會係啱。

## 更多例子

• ${\displaystyle x^{5}-2x^{3}-3=0}$  嘅值係一定少過 ${\displaystyle 2}$
• 對所以嘅整數 ${\displaystyle x}$ ${\displaystyle y}$ ，如果 ${\displaystyle xy}$  係單數，咁 ${\displaystyle x}$ ${\displaystyle y}$  都係單數。

## 參考

• Beth, E. W. (1970). Proof by Contradiction. In Aspect of Modern Logic (pp. 30-41). Springer Netherlands.
• Antonini, S., & Mariotti, M. A. (2006). Reasoning in an absurd world: difficulties with proof by contradiction. In Proceedings of the 30th PME Conference, Prague, Czech Republic (Vol. 2, pp. 65-72).