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Take the example of rolling one $6$ sided dice. If an event $A$ represents a null set, what would be the outcome or the result of getting the event of an empty set?

For example if $A = \{1,2,7\}$ you can say that $A$ is the set of an outcome of $1, 2,$ or $7$ which the last is impossible. Or $A$ could be the sample space meaning the set of all outcomes, but if $A$ is the null set, does it mean that there are no outcomes, i.e. the dice was not rolled?

I was trying to find independent and not disjoint events for this die example. This led me to consider the null set, but I couldn't visualize what the meaning of the null set was for this scenario.

Air Mike
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JobHunter69
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  • See whether this helps you: http://math.stackexchange.com/questions/1255726/what-is-an-empty-set –  Feb 03 '17 at 03:35
  • @Rohan I'm reading that right now, but it doesn't really seem to address the outcome of a physical event – JobHunter69 Feb 03 '17 at 03:35

2 Answers2

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I think it helps to view events as payoff sets in games. For $A=\{1, 2, 7\}$, for example, we play the game where you roll a $6$-sided die, and I win if it comes up $1, 2$, or $7$.

The empty set, then, corresponds to the "fixed game": you roll a $6$-sided die, and I lose.

There are other "fixed game" events (that is, events with zero probability): e.g. $\{7\}$. But those are not obviously fixed (maybe one of the faces of the die is marked $7$! Or maybe I can't count very well!), while the empty set corresponds to a game which is fixed by definition.

Noah Schweber
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  • Yes that seems like a better way of viewing things, but then we aren't treating outcomes as outcomes? – JobHunter69 Feb 03 '17 at 03:42
  • @Goldname "We aren't treating outcomes as outcomes" - I'm not quite sure what you mean. In what way aren't we treating outcomes as outcomes? The game apparatus is just intuitive "sugar" on top of the mathematics, it isn't going to get in the way of anything else we do in probability. (Imagine some process with outcomes going on, and several thousand miles away some silly people with really good telescopes betting on it - the game they play can describe the situation going on, but clearly doesn't interfere with other interpretations of it.) – Noah Schweber Feb 03 '17 at 03:44
  • I mean, the outcome itself is getting the number on the die, but saying that I lose or win if I get a 4 means that me losing or winning is the outcome – JobHunter69 Feb 03 '17 at 03:47
  • @Goldname I guess I don't think of it as changing the outcome - either it's a rephrasing of the same outcome, or it's the outcome of a different event (which happens to be dependent on - in fact, equivalent to - the event you care about). – Noah Schweber Feb 03 '17 at 03:51
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This is very similar to the physical significance of any number.

I'll use $0$ as an example since it's more interesting from a physical perspective than the others. While certainly, negative numbers have no physical relevance.

If you just say $0$, it has no physical relevance. You can have $0$ chairs, $0$ cars, $0$ snow flakes, etc. But that depends entirely on us to give it meaning, and as long as we don't we can argue (equally validly) that either it has no physical meaning or that it has an infinite number of possible meanings which are not yet determined.

I guess a lot of pure math is like that.

So a null set is not a null set in relation to anything as long as there is no context.

To answer your question specifically, in the case of rolling dice, we could agree that this simply means the dice has not been rolled yet. Though, this interpretation is in no way fundamental or essential.

ThisIsNotAnId
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  • "You can have $0$ chairs, $0$ cars, $0$ snow flakes, etc." That's also true of $1$, $17$, or really any number; so how is it relevant here? And "the dice haven't been rolled yet" is a really misleading interpretation of $\emptyset$ as an outcome in this context - the point is that the event (dice rolling) does happen, it's just that $\emptyset$ corresponds to an outcome that can't possibly occur. Now, while it's true that "the dice have not been rolled" is an outcome that can't possibly occur after you roll the dice, I still think that's a misleading choice of interpretation to make. – Noah Schweber Feb 03 '17 at 04:04
  • I understand your point. I was just trying to give a more accessible view of what it means to work with 0 and the null set from an entirely physical perspective. We know these objects are relevant on a piece of paper, but numbers in physical reality come into existence solely with something being there to justify their existence. That's why we can look at a car and say $1$ car. While on the other hand looking around we would not be justified in simply claiming $0$ (or $1$, etc.) unless we also went out of our way to specify $0$ of what. I also had a similar approach in mind for the null set. – ThisIsNotAnId Feb 03 '17 at 04:11
  • @NoahSchweber I also want to mention this question and our discussion comes very close to metaphysical philosophy and what it means to exist. For example, I can argue that not a single car exists. Heck, I can argue that nothing exists. – ThisIsNotAnId Feb 03 '17 at 04:15
  • Also, for people downvoting, could you please share your concerns? – ThisIsNotAnId Feb 03 '17 at 04:18