Proof of 1 = 0, use of ill-formed statements

Question

In his book "Analysis 1", Terence Tao writes:

A logical argument should not contain any ill-formed statements, thus for instance if an argument uses a statement such as x/y = z, it needs to first ensure that y is not equal to zero. Many purported proofs of “0=1” or other false statements rely on overlooking this “statements must be well-formed” criterion.

Can you give an example of such a proof of "0=1"?

Answer 1

x = y.

Then$ x^2 = xy$

Subtract the same thing from both sides:

$x^2- y^2= xy - y^2$

Dividing by (x-y), obtain

x + y = y.

Since x = y, we see that

2 y = y.

Thus 2 = 1, since we started with y nonzero.

Subtracting 1 from both sides,

1 = 0.

Answer 2

Start with the assumption $$x = 0$$ Divide both sides by $x$ to get $$x/x=0/x$$ and thus $$1=0$$

That's the general scheme. Of course it generally gets more obfuscated, for example by starting with the assumption $a+b=c$ and then later dividing both sides with $c-a-b$.