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From what I understand, if A is a set, then B is a subset of A if and only if all the members of B are also members of A. (B ⊆ A)

However, I have come across two questions whose solutions I can't understand

{b,c}⊆{{a,b},{b,c},{a,c}} (False)

Why this is false when it appears that {b,c} is contained within the set {{a,b},{b,c},{a,c}}?

Similiarly, I understand that membership as: A∈B if B is a set and A belongs to it

{a,b,c} ∈ {b,c,a} (False)

I don't understand why this is false, as all the members of A (a, b and c) appear to be members of B (b, c, a)?

I would appreciate any help in clarifying this, thank you

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  • $\begingroup$ Can you specify where these statements are shown as false ? $\endgroup$ – 0xVikas Mar 6 '18 at 7:37
  • $\begingroup$ It's a bit of a semantic thing. The members of the set $\{b, c\}$ are the individual elements $b$ and $c$, but the members of that larger set aren't individual elements, but rather sets of elements. Specifically, that larger set has the set $\{b, c\}$ as an member, not the individual elements $b$ and $c$ themselves. This distinction arises because the set-theoretic axioms (typically those outlined in $\text{ZFC}$ unless otherwise stated) allows for sets to contain other sets. $\endgroup$ – Kaj Hansen Mar 6 '18 at 7:49
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{b,c} is not a subset of {{a,b},{b,c},{a,c}} because {{a,b},{b,c},{a,c}} does not contain the elements b and c. In the same way, {b,c,a} does not contain the whole set {a,b,c} so it doesn't belong to the former set. It would be true if the statement was : {a,b,c} ∈ {b,c,a,{a,b,c}}

Hope it clears your doubt.

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One must be careful to distinguish between "contains as an element" and "contains as a subset".

An object is an element of a set of that element exists within the set. This means that the single object appears in the specification of the set, without being surrounded by braces.

A set $S$ is a subset of a given set $X$ of every element of $S$ is an element of $X$.

In your case, it is true that the set $\{b,c\}$ is an element but it is not a subset, since neither $b$ nor $c$ appear as elements of the right hand side.

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  • $\begingroup$ So would it be right to say that {b,c}∈ {{a,b},{b,c},{a,c}} would be true because {b,c} is an element of {{a,b},{b,c},{a,c}} (i.e. a member but not a subset?) $\endgroup$ – smiles47 Mar 6 '18 at 7:53
  • $\begingroup$ That's correct. the set $\{b,c\}$ is an element of $\{\{a,b\},\{b,c\},\{a,c\}\}$ because it appears (exactly) in the right hand side set. However, for $\{b,c\}$ to be a subset, both $b$ and $c$ would have to appear (individually) without an surrounding braces. $\endgroup$ – SamM Mar 6 '18 at 8:02
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Intuitive idea: a set is a unordered list of elements.

$x\in A$ means "$x$ is an element of $A$" ($x$ is in the list).

In your example, $\{b,c\}\in\{\{a,b\},\mathbf{\{b,c\}},\{a,c\}\}$ is true because $\{b,c\}$ is in the list.

$B\subseteq A$ means $x\in B\implies x\in A$.

In your example, $\{b,c\}\subseteq\{\{a,b\},\{b,c\},\{a,c\}\}$ is false because $b\not\in\{\{a,b\},\{b,c\},\{a,c\}\}$ and $c\not\in\{\{a,b\},\{b,c\},\{a,c\}\}$ (failing in one case is enough).

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