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