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Very simple question, I prob look like an idiot lol. but hey, I'm here to learn

so if I have a set A where A = {$a∈ℕ$ | $a^2 = 5$}

is set A then A = ∅ or is it A = {∅}?

also, if it's the latter, if I have set B say B = {1, 2, 3} and want to find the union of A and B, is it {1, 2, 3} or {∅, 1, 2, 3}? and what about the intersection? if A = {∅, 1, 2} and B = {1, 2, 3}, then A n B = {1, 2} right?

Asaf Karagila
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CurryFlurry
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2 Answers2

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Hint: If $x$ is any element of $\{\,a\in\Bbb N\mid a^2=5\,\}$, then both $x\in \Bbb N$ and $x^2=5$. Soooo, is either of these conditions true when $x=\emptyset$?

Hagen von Eitzen
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  • oh so it's the previous.. thanks :) – CurryFlurry Oct 28 '21 at 20:17
  • You’ve activated my trap card (which is that natural numbers can be represented as Von Neumann ordinals, so in this representation it’s accurate to say $\emptyset \in \mathbb{N}$, but also that this is totally irrelevant to the OP). – Mark Saving Oct 28 '21 at 20:36
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This got a bit long so I will write it here. One way of seeing the notation of sets is what is "in" the curly bracket and the empty set can also be written as $\{\}$ which is notation that is seen. The empty set while not containing anything is itself a thing the same way and empty box is a thing. The set $\{\emptyset\}$ is not the empty set the same way a box containing an empty box is not empty. Second notice $A$ must contain natural numbers or rather it is a subset of the natural numbers on the $\{\emptyset\}=\{\{\}\}$ is a set containing the empty set. Unless you are working with a particular construction of natural numbers done in axiomatic set theory the empty set is not a natural number so the set $\{\emptyset\}$ cannot be $A$ simply because it contains an non natural number element

MIO
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