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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  
Senior Member Joined: Aug 2012 Posts: 2,010 Thanks: 574  Quote:
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 welldefined 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  
Senior Member Joined: Jun 2015 From: England Posts: 853 Thanks: 258  Quote:
There is a difference between the number zero and the term null in mathematics. Last edited by skipjack; July 15th, 2018 at 12:43 PM.  
July 15th, 2018, 11:40 AM  #7 
Senior Member Joined: Aug 2017 From: United Kingdom Posts: 234 Thanks: 78 Math Focus: Algebraic Number Theory, Arithmetic Geometry  
July 16th, 2018, 09:27 AM  #8 
Math Team Joined: Jan 2015 From: Alabama Posts: 3,261 Thanks: 894  

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