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I saw an answer for this question but I don’t understand yet, but what I concluded from it, is hat if $A \subset B$ then every element from $A$ is an element from $B$ but there are some elements in $B$ that are not in $A$, which means $B$ is bigger than $A$. Is it true?

And if $A \subseteq B$ so every element from both sets belong to the other one, which means always $A=B$ and it’s the same for $B \subseteq A$ ($A \subseteq B = B \subseteq A$).

Am I right? And please if there are any addition I would like to hear it.

Thanks!

Rócherz
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utsu
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  • See here. – Dietrich Burde Nov 03 '23 at 10:40
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    Note the similarity (in notation and meaning) used for inequalities: $x<y$ and $x\le y$. – DominikS Nov 03 '23 at 10:43
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    @DominikS For inequalities there is an agreement, but not for sets - see the link I have given. – Dietrich Burde Nov 03 '23 at 10:45
  • $A\subseteq B$ means that either $A\subset B$ or $A=B$. It simply allows for the additional possibility that $A=B$. But as @DietrichBurde points out, sometimes $\subset$ is used instead of $\subseteq$, which adds to the confusion. – DominikS Nov 03 '23 at 10:45
  • @DietrichBurde: Yes, but this does not appear to be the issue in the question. – DominikS Nov 03 '23 at 10:46
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    If you say $A \subset B$ then $A$ is a subset of $B$ and every element of $A$ is an element of $B$. Some people (not me or others) would also say that it means $A$ is a proper subset of $B$ so there is at least one element of $B$ which is not an element of $A$; they would then use $A \subseteq B$ to mean $A$ is a subset of $B$ or $A$ is equal to $B$ (I would use $A \subset B$ for that, and $A \subsetneq B$ for a proper subset). If you really want to avoid using an ambiguous $A \subset B$, you can use $A \subseteq B$ and $A \subsetneq B$ for the two cases, but this might be seen as pedantic. – Henry Nov 03 '23 at 10:47
  • As for "$B$ is bigger than $A$", the even integers are a proper subset of the integers but, in terms of cardinality, the two are the same size. – Henry Nov 03 '23 at 10:50

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$A⊂B$ is used by many authors to imply $A$ is a proper (or strict) subset of $B$, meaning that every element of $A$ is also an element of $B$, but that $A$ cannot be the same as $B$, i.e. there must exist at least one element in $B$ which is not in $A$.

$A⊆B$ implies $A$ is a subset of $B$, but may or may not be a proper subset. Hence every element of $A$ is in $B$, but there may or may not be elements in $B$ that are not in $A$.

To show two sets have exactly the same elements will be represented as $A=B$, which implies $A⊆B$ and implies $A⊂B$ is false. However, $A⊆B$ does not imply $A=B$, as if $A$ is a proper subset of $B$ then $A⊆B$ holds but $A=B$ does not.

JuvHuffpuff
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    Of course, other authors use $A⊂B$ to instead imply that $A$ is a subset of $B$, whether proper or otherwise. – JuvHuffpuff Nov 03 '23 at 10:47
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    I would use $A \subsetneq B$ if I wanted to insist that $A$ is proper, so just $A \subset B$ I would be cautious about assuming either way. – Uzai Nov 03 '23 at 10:51