Proving that the notion of difference of cardinalities is not well-defined.

Question

Consider the following operation on cardinalities. Given sets A,B write |A|-|B|=|A-B|.

Prove that this notion of difference of cardinalities is not well-defined.

Proof:

To Prove that |A|-|B|=|A-B| is not well-defined we will give counter example.

To begin with;

Let A={a,b,c,d,e} and B={h,i,j}

From the above, it is clearly seen that |A|=5 and |B|=3

If we consider the Left Hand Side:

|A|-|B|=5

Now consider the Right Hand Side:

|A-B|=2

Since there is no bijection between |A|-|B| and |A-B|, it is then concluded that the notion of difference of cardinalities is not well-defined.

Can anyone correct me on this please!!!

Answer 1

To show that the notion given is not well defined you have to do the following: Find four sets $A,B,C,D$ such that $|A|=|B|$ and $|C|=|D|$ while $|A|-|C|\ne |B|-|D|$. In your answer above you did not do that. Moreover, your claim that "there is no bijection between $|A|-|B|$ and $|A-B|$" is wrong since the notion of difference you are considering says explicitly that $|A|-|B|$ is $|A-B|$.

You need to work with the notion of equality of cardinalities whereby $|X|=|Y|$ precisely when there exists a bijection $X\to Y$, and not rely on your existing intuition regarding counting finite sets. Again, you can present your counter example by exhibiting four sets as I explained above.

EDIT: To clarify what I mean then: A source of confusion about such questions is to think that one shows the notion not to be well-defined if one shows that notion does not agree with ordinary counting. That is a wrong conclusion though and if one sticks to the precise definitions of cardinalities instead of reverting to counting then one avoids this pitfall. The reason the notion is ill-defined is not it's weird computational results but their inconsistency. For instance, defining $|A|-|B|=|\emptyset |$ is a well-defined notion. It gives weird results that to us have nothing to do with subtraction but it is a well-defined notion. The notion in the question though is not well-defined. Totally different.