Abstract math, sets and logic proof

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Homework Statement


If A is a set that contains a finite number of elements, we say A is a finite set. If
A is a finite set, we write |A| to denote the number of elements in the set A. We
also write |B| < ∞ to indicate that B is a finite set. Denote the sets X and Y by
X = {T : T is a proper subset of P(Z) or |T| < ∞}; Y = {T element of X : T≠ ∅}
Prove or disprove the following:
(there exist X element of R)(∅ element of R and ( for all S element of Y)(|R|≤ |S|}


Homework Equations




The Attempt at a Solution


I think that statement is true because of or in the statement, but I have no idea how to prove it
 
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I can't understand what it is that you are trying to show. Can you write it out in words?
 
It s number 5 from the attachment.
 

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Thread 'Use greedy vertex coloring algorithm to prove the upper bound of χ'
Hi! I am struggling with the exercise I mentioned under "Homework statement". The exercise is about a specific "greedy vertex coloring algorithm". One definition (which matches what my book uses) can be found here: https://people.cs.uchicago.edu/~laci/HANDOUTS/greedycoloring.pdf Here is also a screenshot of the relevant parts of the linked PDF, i.e. the def. of the algorithm: Sadly I don't have much to show as far as a solution attempt goes, as I am stuck on how to proceed. I thought...
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