# Given a set and an element, can we say that the element is subset of the given set? [duplicate]

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Let us consider a set $A = \{1, 2, 3, 4, 5\}$ and an element say $1$.

Can we say that $1$ is subset of set $A$.

If not, please explain.

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• A set of which every element is a subset is called transitive. Unsurprisingly for something we bothered naming, transitivity is a property only of some sets. If you want to construe natural numbers as subsets of natural transitive sets, look up how we model naturals as ordinals. – J.G. Jul 24 '18 at 12:01
• Formally it depends on how $1$ is defined. If e.g. $1:=\{2\}$ (a far-fetched choice) then element $1$ is also a subset of $A$. – drhab Jul 24 '18 at 13:21
• ...and $\{1\}$ is both an element and a subset of $\{1,\{1\},\dots\}$ – AccidentalFourierTransform Jul 24 '18 at 13:38

## 3 Answers

No, we say that

• $1$ is an element of $A$, that is $1\in A$

and

• $\{1\}$ is a subset of $A$, that is $\{1\}\subseteq A$

$1$ can not be subset of $A$ because $1$ is not a set. We can say that a set $B$ is a subset of a set $A$ if every element of B is also element of A. For example, the sets: $\{1, 2\}, \{2, 4, 5\}$ are subsets of A.

Unless you explicitly define another set which contains the element 1, then we cannot say that 1 is the subset of A.

There are exceptions, however.

Let's consider an rather unlikely, but still very much possible, scenario in which we define the symbol '1' as a set with elements {2,5}. In this case, then 1 is definitely a subset of A, because 1 now a symbol which denotes a set that contains the elements which also belongs to A (2 and 5).