What is the flaw in this logic?

Question

I have never taken set theory (so go easy on me), I know only what I have read.

Firstly, is there a problem with this statement?

If there exist sets A and B, such that: |A| = |B|, then

|(A\B) + (A⋂B)| = |(B\A) + (A⋂B)|

|(A\B)| + |(A⋂B)| = |(B\A)| + |(A⋂B)|

|(A\B)| = |(B\A)|

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If the above is true, then how does it apply to infinite sets and bijection? i.e. it's cited that |N| = |E| for cardinality of natural numbers and even numbers.

So if we assume |N| = |E|, then

|O + E| = |{} + E|

|O| + |E| = |{}| + |E|

|O| = |{}|

No. Why not?

Answer 1

The cancellation law does not hold for cardinal addition. The sum of two equal infinite cardinals is the same as one of them.

Answer 2

Your first and second equations are correct: the third is only necessarily true for finite sets.

The thing is, you cannot subtract infinite cardinals. If $C,D$ are two sets of infinite cardinality, then: $$|C|+|D|=\max(|C|,|D|)$$ Or so I am told. In your situation, $|A\setminus B|+|A\cap B|=|B\setminus A|+|A\cap B|$ does not imply $|A\cap B|$ can be validly cancelled from both sides: the maximum function can’t be inverted like that!

In particular: $$|O|+|E|=0+|E|$$Is true. $|O|=0$ is false, and the flaw in the logic is that you can’t ‘add $-|E|$ to both sides’.