Take for example the collection $X=\a, b\$. I don"t see $\emptyset$ anywhere in $X$, therefore how have the right to it it is in a subset?

$\begingroup$ "Subset of" means something different than "element of". Keep in mind $\a\$ is additionally a subset that $X$, despite $\ a \$ not showing up "in" $X$. $\endgroup$
that"s since there room statements that space vacuously true. $Y\subseteq X$ means for every $y\in Y$, we have $y\in X$. Now is the true the for every $y\in \emptyset$, we have actually $y\in X$? Yes, the explain is vacuously true, since you can"t pick any $y\in\emptyset$.

You are watching: The empty set is a subset of every set

Because every solitary element that $\emptyset$ is additionally an element of $X$. Or can you surname an element of $\emptyset$ that is no an aspect of $X$?

You have to start native the definition :

$Y \subseteq X$ iff $\forall x (x \in Y \rightarrow x \in X)$.

Then friend "check" this meaning with $\emptyset$ in location of $Y$ :

$\emptyset \subseteq X$ iff $\forall x (x \in \emptyset \rightarrow x \in X)$.

Now you should use the truth-table meaning of $\rightarrow$ ; you have actually that :

"if $p$ is false, then $p \rightarrow q$ is true", for $q$ whatever;

so, due to the fact that :

$x \in \emptyset$

is not true, for every $x$, the above truth-definition that $\rightarrow$ provides us that :

"for all $x$, $x \in \emptyset \rightarrow x \in X$ is true", for $X$ whatever.

This is the reason why the emptyset ($\emptyset$) is a subset of every collection $X$.

See more: The Maize At Little Darby Creek In, The Maize At Little Darby Creek

re-publishing
point out
monitor
edited Jun 25 "19 at 13:51
answered january 29 "14 at 21:55

Mauro ALLEGRANZAMauro ALLEGRANZA
$\endgroup$
1
4
$\begingroup$
Subsets are not have to elements. The facets of $\a,b\$ are $a$ and also $b$. But $\in$ and $\subseteq$ are different things.

re-publishing
cite
monitor
answered january 29 "14 in ~ 19:04

Asaf Karagila♦Asaf Karagila
$\endgroup$
0
include a comment |

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

Featured ~ above Meta
20
Is the null set a subset the every set?
0
Is this proof correct? If not, whereby is the flaw?
0
Set theory; sets and also subsets; Is one empty collection contained in ~ a collection that consists of real numbers?
0
Any collection A has actually void set as that subset? if correctly how?
associated
10
straight proof the empty collection being subset of every set
3
If the empty set is a subset that every set, why isn't $\\emptyset,\a\\=\\a\\$?
1
A power set contais a set of a empty subset?
3
How deserve to it be that the empty set is a subset that every set but not an facet of every set?
3
Is the set that includes the empty collection ∅ likewise a subset of every sets?
hot Network questions much more hot inquiries
jamesmerse.comematics
agency
ridge Exchange Network
site design / logo design © 2021 stack Exchange Inc; user contributions licensed under cc by-sa. Rev2021.9.16.40224

jamesmerse.comematics ridge Exchange works finest with JavaScript enabled

her privacy

By click “Accept every cookies”, you agree ridge Exchange can store cookies on your maker and disclose info in accordance v our Cookie Policy.