So I was solving a problem in Rudin (chapter 3 #16, to be specific) and I realized how convenient it would be to have a symbol that represented an undetermined equivalence relationship. As an example I will use the symbol $\sim$.
$\mathbf{Example}$. Suppose we have an expression $A$ that we want to relate to $B$. We then set $$ A \sim B $$ and perform an algebraic operation to obtain $A'$ and $B'$. If this operation involved multiplying both sides by a negative number, we change $\sim$ to $\sim'$. So suppose it did; then we have $$ A' \sim' B'. $$ Lets perform another algebraic operation, and suppose again that we multiplied by a negative number. Then $\sim'$ becomes $\sim$ (since inequalities are the same under two multiplications by negatives), and we obtain $A''$ and $B''$ and thus $$ A'' \sim B''. $$ We can continue on in this fashion until we have have done $n$ operations. For simplicity let us suppose that at this point our equivalence relation is $\sim'$. Then $$ A^{(n)} \sim' B^{(n)}. $$ Suppose further that we actually know that $A^{(n)} < B^{(n)}$. Then we can conclude $$ A > B. $$ If instead we knew that $A^{(n)} = B^{(n)}$ we would have $A = B$, and if instead we knew that $A^{(n)} > B^{(n)}$ we would have $A < B$.
Does anyone know of a symbol such as this, and if so, are there interesting things to be said about essentially solving for equivalence relations?
