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March 23rd, 2018, 11:34 PM  #1 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3  group theory question
Singleton is a group that has only one item. For example, {0} is singleton. In group theory, the number 1 is defined as singleton {0}. What's the meaning of the sentence above: in group theory, the number 1 is defined as singleton {0}. and Why? Last edited by skipjack; March 24th, 2018 at 04:26 AM. 
March 24th, 2018, 04:41 AM  #2 
Global Moderator Joined: Dec 2006 Posts: 21,127 Thanks: 2336 
Which should be defined first, the number 1 or the adjective one?

March 24th, 2018, 05:02 AM  #3 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
Tell me, what the different? I write the statement as It Wrote... "In group theory, the number 1 is defined as singleton [of the group =] {0}. 
March 24th, 2018, 05:28 AM  #4 
Global Moderator Joined: Dec 2006 Posts: 21,127 Thanks: 2336 
For the number 1 to be defined as the singleton {0}, the adjective one needs to have been defined first, because the definition of a singleton uses the adjective one.

March 24th, 2018, 05:38 AM  #5 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
I don't understand the reply of you. Can you please elaborate and use another word than the word adjective or explain it mathematical meaning. Can you write an answer to my question? Thank... [Please elaborate] Last edited by policer; March 24th, 2018 at 05:56 AM. 
March 24th, 2018, 06:10 AM  #6  
Senior Member Joined: Oct 2009 Posts: 912 Thanks: 354  Quote:
Here is the definition of the singleton 0 in standard ZFC set theory: A = {0} iff $$(0\in A)~\text{and}~\forall y: (y\in A\rightarrow y=0)$$ Nowhere is the adjective one used...  
March 24th, 2018, 06:51 AM  #7 
Global Moderator Joined: Dec 2006 Posts: 21,127 Thanks: 2336 
I was referring to the definition in policer's post, not the standard definition.

March 24th, 2018, 07:17 AM  #8 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
I think I wander between two definition. What the meaning of adjective one? Last edited by skipjack; March 24th, 2018 at 10:29 AM. 
March 24th, 2018, 08:45 AM  #9 
Global Moderator Joined: Dec 2006 Posts: 21,127 Thanks: 2336 
Although policer's original post used the term "group", the example given was {0}, which is a set (as a group operation isn't specified). This example suggests that 0 is already defined, so let's define 0 as {} (the empty set). What about the definition of a singleton? That used the term one, but, as Micrm@ss showed, the definition of a singleton can be accomplished in a way that avoids use of one. However, a definition of 1 could be useful anyway, so let's define it as {0} (the set just used as an example of a singleton). Now what can be said about 1 that's not true of 0?

March 24th, 2018, 09:12 AM  #10 
Banned Camp Joined: Dec 2017 From: Tel Aviv Posts: 87 Thanks: 3 
Thanks, now It clear.


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