圖畫10 數學家的『本性』——鑽牛角尖？

Discrete Mathematics:
Give a proof by contradiction for the following statement.


There is no integer that is both even and odd.

(Of course we know this ==)


Proof: (What?!)


[We take the negation of the statement and suppose it to be true.]
(Isn't it true already? ==)


Suppose there is an integer x that is both even and odd.
(wookay... then?)


By definition of even numbers,

x=2k1, where k1 is an integer

By definition of odd numbers,

x=2k2+1, where k2 is an integer

(But then both are different wor zzz...)

2k1=2k2+1

(k1+k2)=1/2

(k1+k2) is an integer but 1/2 is not an integer.

A subtraction of integer numbers should be integer number.

--> Contradiction happens.

(Like this also can?!)

Therefore there is no integer that is both even and odd.

(... ... ... ... ... ... ... ... ... ...)

This is just one of the sample that is easier to understand by everyone. We had more tak-tau-apa-macam-proof contradictions last semester in Real Analysis. ==