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If $A$ & $B$ are two sets, what does '$A$ is contained in $B$' mean ? Does it mean that $A$ is a subset of $B$ or $A$ is a proper subset of $B$?

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  • $\begingroup$ There is no universal convention, check the one used by the particular author. $\endgroup$ – Yves Daoust Jan 12 at 15:28
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It's a bit ambiguous, but hopefully it's only used when there is little chance of confusion. If there is a chance, it is better to use notation. Either $A\in B$ for $A$ being an element of $B$, or $A\subseteq B$ for $A$ being a subset of $B$.

If it's the latter, generally it is not necessarily a proper subset. Authors however can do whatever they want, as long as they define it. You have to watch out for nonstandard language and notation.

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    $\begingroup$ @Thomas Started writing it on my phone, had to do something else, must've accidentally posted it without finishing. Thanks. $\endgroup$ – Matt Samuel Jan 12 at 15:27
  • $\begingroup$ I would go as far as saying that writing "$A$ is contained in $B$" and meaning $A\in B$ amounts to gaslighting... $\endgroup$ – JuliusL33t Jan 12 at 15:56
  • $\begingroup$ @Julius Sets can contain elements. And in set theory everything is a set. So in situations like that it can legitimately be ambiguous. $\endgroup$ – Matt Samuel Jan 12 at 16:19
  • $\begingroup$ I'm not commenting on the mathematical content, but on pedagogy and custom. Some people define "cat" to mean "dog" and "up" to mean "down". Saying "$A$ is contained in $B$" and meaning "$A$ is an element of $B$" seems so cruel to me, especially when this is very standard terminology. But hey, whatever floats your boat. $\endgroup$ – JuliusL33t Jan 12 at 16:36
  • $\begingroup$ I DO NOT disagree with you answer, so don't take offense. $\endgroup$ – JuliusL33t Jan 12 at 16:39
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Because we do know nothing about the two sets let me give you two examples.

1) We know that $N \subset R$. This means that $N$ is contained in $R$.

2) Let us define $A = \{x,y\}$ and $B = \{x,y\}$ then A is contained in B and B is contained in A.

To answer your question, both of them can be correct.

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The only thing that "$A$ is contained in $B$" means is that the implication

$$x \in A \implies x \in B $$

is satisfied. It could be a proper or non-proper subset. Often, when the subset is proper, most authors explicitely state this.

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