Can there exists three sets: $A \subset B$, $B \in C$, $A \in C$

Question

Can there exists three sets: $A \subset B$, $B \in C$, $A \in C$, if not, why not?, if yes, give an example.

My example:

A={$a,b$}

B={$a,b,c$}

C={{$a,b$},{$a,b,c$}}

Is this all? It seems this is a tricky question and there's something I'm missing. Thank you for your help!

Edit: fix typo

Except for one "}" you forgot, yes. If xour $\subset$ stands for $\subseteq$ and not $\subsetneq$, you could even use $A=B=\emptyset$, $C={\emptyset}$. — Hagen von Eitzen, Dec 15 '19 at 06:42
Kuratowski's version of ordered pairs is $(a,b):={{a},{a,b}}$ and this gives another example of what you want with $A={a}$, $B={a,b}$ and $C=(a,b)$. — მამუკა ჯიბლაძე, Dec 15 '19 at 07:18

score 1 · Answer 1 · 2019-12-15T08:49:51.993

1

If A if a subset of B and If B is an element or subset of C

Then A can only be a subset/element of C.

Thanks for clarifying. i misinterpreted the relationship b/w B and C.

edited Dec 15 '19 at 08:49

answered Dec 15 '19 at 08:20

B is an element of C, not a subset of C ( $B \in C$) – Belen Dec 15 '19 at 08:37

score 1 · Accepted Answer · answered Dec 15 '19 at 08:48

Written down in plain text, your question reduces to the existence of two sets, one of which a proper subset of the other such that they are both elements of a third set! There are infinitely many instances of triplets of sets satisfying such a configuration: for any set $A$ there must exist an object $a$ not an element of it (otherwise $A$ would become a universal set with respect to the relation of set-theoretical membership, situation which is prohibited by the axioms of almost any modern axiomatic system for formal mathematics); therefore, by setting $B=A \cup \{a\}$ you automatically have $A \subset B$; furthermore, by considering $C=\{A, B\}$ you will have obtained a triplet of sets as specified starting from any given set.

score 0 · Answer 3 · answered Dec 15 '19 at 08:40

0

My example given above is correct, therefore

$$\exists A,B,C \hookrightarrow A \subset B, A \in C, B \in C $$

answered Dec 15 '19 at 08:40

Belen

568

Can there exists three sets: $A \subset B$, $B \in C$, $A \in C$

3 Answers3