We understand that the empty collection Ø is always a subset of every set, however is null Ø always an element of every set as well?
Ø = is the empty set, with no elements. Ø is not the north set, it's a set containing 1 element which is the empty set, Ø. Ø is subset of any set, yet Ø isn't necessarily an aspect of a set. For example Ø isn't an element of Ø, since Ø has no elements.
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Hello! I'm additionally learning math. I was right about OP's question, yet your answer gave me another: have the right to Ø it is in an facet of a collection of number or objects, due to the fact that Ø chin is a set? Is over there a difference between a collection of objects and also a collection of sets, or can a collection contain whichever with no problems?
I mean, ns guess a set could be 3, dog, 4, 5, 6, Napoleon, yet can a collection be characterized to have only one kind of element, without specifying what the is? I'm not expressing myself plainly at all. I mean, is it advantageous to to speak "a collection of integers" (could not contain Ø together an element) vs "a collection of sets" (which could)?
...This isn't coming the end right. Feel free to effort to read my mind and figure out what I'm trying come ask.
Ø is no an facet of every set. Because that example, the is no an aspect of itself, due to the fact that Ø has no elements.
In fact, many sets you job-related with don't have Ø together an element. Because that example, 1,2,3 go not have Ø as an element.
The empty collection = is a subset of any type of set, due to the fact that every facet in the empty set is in every set, yet is no an facet of every set. It's a ethereal difference, but all you should do is job-related through the interpretations rigorously.
I reckon you typical to questioning what is a collection ultimately do of? If so below is solution i composed as reply to who here:
The axioms that ZFC execute not say much around what is a set. You deserve to however add further axioms come ZFC such the every collection is do of the north sets to placed it intuitively. So to speak the set dog that you provided in her example collection would watch sth like dog= empty, empty, empty. This is referred to as the axiom of regularity.
Here is an additional interesting bit. We specify the organic numbers in set theory through letting empty set stand for 0, 1 is empty and define x+1 as x U x. For this reason 1=0+1 = empty U empty = empty. From over there we can define addition and multiplication recursively. So the herbal numbers in collection theory room nice in the sense that us didnt use an ext than the empty set to create them (and ofc the axioms).
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If friend study collection theory deep enough you will have actually a far better grasp that what ns am saying. Its also so much fun.