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 July 1st, 2018, 10:17 PM #1 Senior Member   Joined: Nov 2011 Posts: 224 Thanks: 2 empty set Is the size of empty set is 0? Can I call the size - power (the power of the set instead of the size of the set)? What the differences between size term to power term?
 July 1st, 2018, 10:35 PM #2 Senior Member   Joined: Feb 2016 From: Australia Posts: 1,662 Thanks: 574 Math Focus: Yet to find out. Yes. After all, it has no elements! Zero, nada, zip, diddly squat... I'm not sure what you are thinking about when you use the term 'power' here. Maybe do some reading: https://en.wikipedia.org/wiki/Empty_set https://en.wikipedia.org/wiki/Cardinality
 July 1st, 2018, 10:52 PM #3 Global Moderator   Joined: Dec 2006 Posts: 19,547 Thanks: 1754 The power set of a set S is the set of all subsets of S. If S is the empty set, its power set has only one element, S. Using the term "power" in other senses might cause confusion. However, it does happen. For example, an exponent is sometimes called a power.
July 1st, 2018, 11:00 PM   #4
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Quote:
 Originally Posted by shaharhada Is the size of empty set is 0? Can I call the size - power (the power of the set instead of the size of the set)? What the differences between size term to power term?
What exactly do you mean by size? The word can be defined a lot of different ways ... cardinality, measure, etc.

The empty set is the set that contains no elements. But what is its size? Its cardinality and its Lebesgue measure are zero. But I will give you an example where it's natural to think of the measure of the empty set as 1/2.

The natural numbers are the set 0, 1, 2, 3, 4, ... as usual. In set theory, we can define what these symbols represent in terms of sets. We define 0 as the empty set, 1 as the set containing 0; 2 as the set containing 0 and 1; 3 as the set containing 0, 1 and 2, and so forth.

Now suppose I want to find a way to pick a natural number "at random." There's no way to do this if we insist that every natural number gets the same probability, but there's a way to do it if we relax that condition.

For a natural number $n$, assign it the probability $\frac{1}{2^{n+1}}$. So 0 has probability 1/2, 1 has probability 1/4, 2 has probability 1/8, and so forth.

Now we see that the sum of all the probabilities is 1, which satisifies the definition of a probability measure; and every natural number has some well-defined probability. And in this scheme, 0 -- or pedantically the empty set -- is assigned size 1/2.

Now this example is a little bit contrived, but it does show that there's a natural context in which we can assign "size" 1/2 to the empty set. And it's still the empty set! It contains no elements. That's the only requirement. Nothing says how big its "size", as long as you can be creative and find an interesting definition of size.

 July 15th, 2018, 06:48 AM #5 Newbie   Joined: Jul 2018 From: UK Posts: 7 Thanks: 0 The empty set is almost its own power set, about as close as you can get. P({}) = {{}} Last edited by skipjack; July 15th, 2018 at 12:44 PM.
July 15th, 2018, 11:15 AM   #6
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 Originally Posted by alan2here The empty set is almost its own power set, about as close as you can get. P({}) = {{}}
Masche has already told you that this is 1 not the null (or empty) set.

There is a difference between the number zero and the term null in mathematics.

Quote:
 Originally Posted by Maschke We define 0 as the empty set, 1 as the set containing 0;

Last edited by skipjack; July 15th, 2018 at 12:43 PM.

July 15th, 2018, 11:40 AM   #7
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Quote:
 Originally Posted by studiot Masche has already told you that this is 1 not the null (or empty) set.
Alan didn't say anything to suggest otherwise.

July 16th, 2018, 09:27 AM   #8
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Quote:
 Originally Posted by studiot Masche has already told you that this is 1 not the null (or empty) set. There is a difference between the number zero and the term null in mathematics.
I was taken aback at first too. But then I noticed that alan2here said "almost"!

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