MHB Write ⊆ or ∈ in the space provided: {Please Check My Solution}

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The discussion focuses on the correct usage of the symbols ⊆ (subset) and ∈ (element) in set theory. The provided solutions indicate that N is an element of ℚ and P(R), while the empty set ∅ is a subset of Z, and √2 is a subset of R. Participants clarify the definitions, stating that a subset contains all elements of another set, while an element is a member of a set. The conversation also prompts a question about whether N is considered a set or an element. Understanding these concepts is crucial for accurately interpreting set relationships.
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Question:
Write ⊆ or ∈ in the space provided:
N _______ ℚ
N _______ P(R)
∅ _______ Z
√ 2 _______ R

Solution:
N ∈ ℚ
N ∈ P(R)
∅ ⊆ Z
√ 2 ⊆ R
 
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Can you state what $\subseteq$ and $\in$ means?
 
MarkFL said:
Can you state what $\subseteq$ and $\in$ means?
⊆ = Subset
∈ = Element

Definition:

If all the elements of a set is contained in ANOTHER set, then the set whose elements are contained in another set is a subset.
Ex. Set A 's elements= 1,2,3 Set B's elements= 1,2,3,4,5 so... that means all the elements of Set A are in Set B, so A ⊆ B.
 
Good! :D

So, in reference to the first part of the question, is $\mathbb{N}$ a set or an element?
 
I was reading documentation about the soundness and completeness of logic formal systems. Consider the following $$\vdash_S \phi$$ where ##S## is the proof-system making part the formal system and ##\phi## is a wff (well formed formula) of the formal language. Note the blank on left of the turnstile symbol ##\vdash_S##, as far as I can tell it actually represents the empty set. So what does it mean ? I guess it actually means ##\phi## is a theorem of the formal system, i.e. there is a...

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