Take for example the set $X={a, b}$. I don”t see $emptyset$ anywhere in $X$, so how can it be a subset?

$egingroup$ “Subset of” means something different than “element of”. Note ${a}$ is also a subset of $X$, despite ${ a }$ not appearing “in” $X$. $endgroup$

Because every single element of $emptyset$ is also an element of $X$. Or can you name an element of $emptyset$ that is not an element of $X$?

that”s because there are statements that are vacuously true. $Ysubseteq X$ means for all $yin Y$, we have $yin X$. Now is it true that for all $yin emptyset $, we have $yin X$? Yes, the statement is vacuously true, since you can”t pick any $yinemptyset$.

You are watching: Is empty set a subset of every set

You must start from the definition :

$Y subseteq X$ iff $forall x (x in Y

ightarrow x in X)$.

Then you “check” this definition with $emptyset$ in place of $Y$ :

$emptyset subseteq X$ iff $forall x (x in emptyset

ightarrow x in X)$.

See more: How To Change Your Picture On 8 Ball Pool Profile Picture (2017)

Now you must use the truth-table definition of $

ightarrow$ ; you have that :

“if $p$ is *false*, then $p

ightarrow q$ is *true*“, for $q$ whatever;

so, due to the fact that :

$x in emptyset$

is **not** *true*, for every $x$, the above truth-definition of $

ightarrow$ gives us that :

“for all $x$, $x in emptyset

ightarrow x in X$ is *true*“, for $X$ whatever.

This is the reason why the *emptyset* ($emptyset$) is a *subset* of every set $X$.

See more: All The Years Of Her Life Summary, All The Years Of Her Life

Share

Cite

Follow

edited Jun 25 “19 at 13:51

answered Jan 29 “14 at 21:55

Mauro ALLEGRANZAMauro ALLEGRANZA

87.4k55 gold badges5656 silver badges130130 bronze badges

$endgroup$

1

Add a comment |

4

$egingroup$

Subsets are not necessarily elements. The elements of ${a,b}$ are $a$ and $b$. But $in$ and $subseteq$ are different things.

Share

Cite

Follow

answered Jan 29 “14 at 19:04

Asaf Karagila♦Asaf Karagila

363k4141 gold badges538538 silver badges921921 bronze badges

$endgroup$

0

Add a comment |

## Not the answer you're looking for? Browse other questions tagged elementary-set-theory examples-counterexamples or ask your own question.

Featured on Meta

Linked

20

Is the null set a subset of every set?

0

Is this proof correct? If not, where is the flaw?

0

Set theory; sets and subsets; Is an empty set contained within a set that contains real numbers?

0

Any set A has void set as its subset? if yes how?

Related

10

Direct proof of empty set being subset of every set

3

If the empty set is a subset of every set, why isn't ${emptyset,{a}}={{a}}$?

1

A power set contais a set of a empty subset?

3

How can it be that the empty set is a subset of every set but not an element of every set?

3

Is the set that contains the empty set {∅} also a subset of all sets?

Hot Network Questions more hot questions

positiveeast.orgematics

Company

Stack Exchange Network

site design / logo © 2021 Stack Exchange Inc; user contributions licensed under cc by-sa. rev2021.11.3.40639

positiveeast.orgematics Stack Exchange works best with JavaScript enabled

Your privacy

By clicking “Accept all cookies”, you agree Stack Exchange can store cookies on your device and disclose information in accordance with our Cookie Policy.

## Discussion about this post