There is only one empty set. The is a subset of every set, consisting of itself. Each set only contains it once as a subset, no an infinite variety of times.

You are watching: Every set is a subset of itself

let $A$ and $B$ be sets. If every element $ain A$ is likewise an facet of $B$, climate $Asubseteq B$.

Flip the around and you get

If $A otsubseteq B$, then there exists some aspect $xin A$ such that $x otin B$.

If $A$ is the empty set, there space no $x$s in $A$, for this reason in details there are no $x$s in $A$ that room not in $B$. For this reason $A otsubseteq B$ can"t it is in true. Furthermore, note that we haven"t used any kind of property of $B$ in the ahead line, so this uses to every collection $B$, consisting of $B=emptyset$.

(From a wider standpoint, you can think of the empty set as the collection for which $xin emptysetimplies P$ is true because that every explain $P$. For example, every $x$ in the empty set is orange; also, every $x$ in the emptyset is no orange. There is no contradiction in one of two people of this statements because there are no $x$"s i beg your pardon could carry out counterexamples.)

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edited Oct 7 "17 in ~ 20:48

answered Mar 20 "13 in ~ 8:45

Alexander Gruber♦Alexander Gruber

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The empty set is subset of the north set, together every element of the empty collection is an facet of the empty set. But $0$ is no in the north set.

$A subseteq B$ when $xin A implies xin B$. Together $xin A iff xin A$ we see that $A subseteq A$ is always true, when $A$ is a set.

A value is a value not a set, sometimes $0$ is defined as the empty set but climate $0$ is the empty collection and not the number.

This wake up for example in category theory, as you are only interested in summary sets, and all set of the same cardinality are in a sense the same, you simply title finite sets by their cardinality.

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edited Mar 19 "13 in ~ 11:32

answered Mar 19 "13 at 11:21

Dominic MichaelisDominic Michaelis

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*Any*collection is a subset the itself. Allow be$$A=x,vert,varphi(x).$$As$$varphi(x)implies varphi(x),$$we have$$Asubseteq A.$$

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answered Dec 4 "14 in ~ 8:19

Martín-Blas Pérez PinillaMartín-Blas Pérez Pinilla

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Because the reflexivity that $subseteq$, for all $A$ collection $A subseteq A$ is true. For $A = emptyset$ we have actually that $emptyset subseteq emptyset$, for this reason the empty collection is a subset of itself.

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answered Dec 7 "14 at 17:09

user153012user153012

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The intersection of 2 sets is a subset of each of the initial sets. So if $phi$ is empty set and A is any set then $ phicap A $ is $phi$ which way $phi$ is subset the A(which is any set) and $phi$ is a subset the $phi$ which suggests empty set is subset the itself.

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answer Mar 8 "16 in ~ 5:53

Ashish PaniAshish Pani

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By the definition of subset this question means:Is every facet of the empty collection a member of the north set?The answer must be yes, because the empty set doesn"t contain any type of elements.

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answered Feb 11 "18 at 5:46

ANorellANorell

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An empty subset have the right to be selected from any set, consisting of the empty set itself, using selection criteria that cannot be satisfied, e.g. Picking those elements $x$ such that $x eq x$.

See mine formal proof (in DC evidence format) at: http://dcproof.com/ExistenseOfNullSet.htm

There is just one empty set. If $phi_1$ and also $phi_2$ are empty sets, then $ phi_1 = phi_2$.

See my formal proof (in DC evidence format) at: http://dcproof.com/UniquenessOfNullSet.htm

**Edit**: for any collection $X$ we can pick an north subset $S$ such that:

$forall a:

or

$forall a:

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edited jan 22 at 4:03

answer Mar 20 "13 at 16:49

Dan ChristensenDan Christensen

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