# Mathematical symbol for “indifference”

Is there any mathematical symbol for the word "indifferent". I mean that if I want to say if $x$ > 0, the machine can indifferently choose between $y$ or $z$. I want to express the last part in a mathematical notation. I know that in economics, we use ~ to denote indifference of preferences. Is there any general mathematical notation for that?

• Do you mean either one or the other must be used or that both can be used and that none has to be used or could both be used as once? – mathreadler Mar 4 '17 at 21:47
• I have always just stated that the choice was arbitrary. There is no mathematical symbol for this as far as I know, though in some cases WLOG (without loss of generality) applies – infinitylord Mar 4 '17 at 21:50
• @mathreadler I meant either y or z can be used without a preference of one over the other. In other words, I want to express that the machine can "randomly" chose between y and z. – salhin Mar 4 '17 at 21:52
• But that it must choose at least one of them and not both at once? – mathreadler Mar 4 '17 at 21:53
• @mathreadler yes – salhin Mar 4 '17 at 21:53

What you seem to want is an xor-decision (exclusive or). This is the logical operation that means "at least one, either one or the other, but not both at once". We can write it as a table

$$\begin{array}{|c|cc|}\hline\oplus&0&1\\\hline0&0&1\\1&1&0\\\hline\end{array}$$

If the number 1 means "chosen" and 0 "not chosen" and

1 in the table means "OK" and 0 in the table means "Not OK".

May seem confusing with the symbols and numbers but in mathematics and computing being a bit extra pedantic is good to know precisely what we are talking about and not risking to screw things up.

If i interpret your question correctly it sounds like you are saying that choosing $y$ or $z$ is equally likely. This corresponds to a discrete uniform distribution over the choices with $p (y)=\frac {1}{2}=p (z)$ because there are 2 choices. With $n$ choices the probability of each choice would be $\frac {1}{n}$. Overall i am interpreting indifference to mean equally likely.

Not directly to your first question, but to your latter remarks: In using $\sim$ to refer to indifference in preference, economists are likely alluding to the fact that indifference is an equivalence relation (i.e. reflexive, transitive, symmetric). Equivalence relations are used to say that various objects are basically the same with respect to some facet we are considering. The same symbol $\sim$ is usually used for equivalence relations.

For your example, you could say "for all $x>0$. The symbol is $\forall x>0$.