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(section:uge1L)=

These exercises are optional extra material for students who are done with the exercises of the day but have the time for and the interest in more.

# Extra (Optional) Exercises -- Short Day

## Extra Exercise 1: "Proofs" ending in Contradictions

+++

### Question a

The following "proof" of the claim $1=0$ is proposed:

$$
\begin{array}{ll}
a=b & \text{The assumption. $a$ and $b$ are assumed real.}\\
 & \text{Multiply by $a$}\\
a^2=ab & \\
 & \text{Subtract $b^2$}\\
a^2-b^2=ab-b^2 & \\
 & \text{Rewrite}\\
(a+b)(a-b)=(a-b)b & \\
 & \text{Divide by $a-b$}\\
a+b=b & \\
 & \text{Use the assumption that $a=b$}\\
2a=a & \\
 & \text{Divide by $a$}\\
2=1 & \\
 & \text{Subtract $1$}\\
1=0 & \\
\end{array}
$$

It's clear that something is wrong - where is the mistake?

%```{hint}
%:class: dropdown
%x
%```
+++

### Question b

Again, a "proof" is presented for the claim $1=0$:

$$
\begin{array}{ll}
-20=-20 & \\
 & \text{Rewrite}\\
25-45=16-36 & \\
 & \text{Rewrite}\\
5^2-9\cdot 5=4^2-9\cdot 4 & \\
 & \text{Rewrite}\\
5^2-9\cdot 5=4^2-9\cdot 4 & \\
 & \text{Rewrite}\\
\left( 5-\frac{9}{2}\right)^2=\left(4-\frac{9}{2}\right)^2 & \\
 & \text{Take the square root}\\
5-\frac{9}{2}=4-\frac{9}{2} & \\
 & \text{Subtract $4-\frac{9}{2}$}\\
1=0 & \\
\end{array}
$$

Where is the mistake this time?

%```{admonition} Svar
%:class: dropdown
%x
%```
%
%---
%
%## Ekstra Opgave 2: