# Membership Uniqueness?

1. Sep 6, 2010

### delphi

Hello All,

I am trying to define a uniqueness of a member of a set, please bear with me as my notation is not as refined as it ought to be:

For a set X:

{ x(i) } union { f(x(j)) = true, where j is not equal to i } = { x(i) }

what I am trying to say is, for this set X there exists only one member whose evaluation with f() results in true.

I am defining this by saying that the union of any element with any other element other than itself in the set who's value is true is equal to the first element, however this doesn't quite work - I almost need a count() function on the right side and say that the count() is always = 1.

Is there a more correct elegant way to say this?

2. Sep 6, 2010

### CompuChip

Yes, you can use the "count" function. The common mathematical notation for this is |{...}| or #{...}:

$$\left| \{ x \in X \mid f(x) = \text{true} \} \right| = 1$$
$$\#\{ x \in X \mid f(x) = \text{true} \} = 1$$

Or you could use quantifiers:
$$\exists! x \in X: f(x) = \text{true}$$
(I don't know if this is official notation, with the ! meaning "unique", but I'm pretty sure any mathematician will understand it).

Or you could write this correct-but-ugly line,
$$\exists x \in X: \left( f(x) = \text{true} \wedge \forall y \in X: y \neq x \implies f(x) \neq \text{true} \right)$$

3. Sep 6, 2010

### SW VandeCarr

It depends on what you mean by "uniqueness". Ir's one thing to know there's exactly one murderer in the room, and another to know which person in the room is the murderer. If you want to know who the murderer is, there needs to be some identifying property or set of properties.

This was a practical problem for me in my work. We solved the problem by creating a family of sets Ai such that each set was identified with a unique property in a way that an intersection of some of the sets contained exactly one member.

Last edited: Sep 6, 2010