$\\begingroup$ \"Subset of\" means something different than \"element of\". Keep in mind $\\a\\$ is likewise a subset that $X$, in spite of $\\ a \\$ not showing up \"in\" $X$. $\\endgroup$

that\"s because there space statements that room vacuously true. $Y\\subseteq X$ way for all $y\\in Y$, we have actually $y\\in X$. Now is it true that for all $y\\in \\emptyset $, we have $y\\in X$? Yes, the statement is vacuously true, because you can\"t pick any kind of $y\\in\\emptyset$.

You are watching: Is the null set a subset of every set

Because every solitary element that $\\emptyset$ is additionally an element of $X$. Or deserve to you surname an element of $\\emptyset$ the is not an facet of $X$?

You must start indigenous the meaning :

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

Then friend \"check\" this an interpretation with $\\emptyset$ in place of $Y$ :

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

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

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

so, due to the reality that :

$x \\in \\emptyset$

is **not** *true*, for every $x$, the above truth-definition of $\\rightarrow$ gives us the :

\"for every $x$, $x \\in \\emptyset \\rightarrow x \\in X$ is *true*\", because that $X$ whatever.

This is the factor why the *emptyset* ($\\emptyset$) is a *subset* the every set $X$.

See more: Larry H. Miller Ford Lakewood

share

mention

follow

edited Jun 25 \"19 in ~ 13:51

answered january 29 \"14 at 21:55

Mauro ALLEGRANZAMauro ALLEGRANZA

87.1k55 gold badges5656 silver badges130130 bronze title

$\\endgroup$

1

include a comment |

4

$\\begingroup$

Subsets room not necessarily elements. The elements of $\\a,b\\$ space $a$ and $b$. Yet $\\in$ and $\\subseteq$ are various things.

re-publishing

cite

follow

answered january 29 \"14 at 19:04

Asaf Karagila♦Asaf Karagila

361k4141 gold badges532532 silver- badges913913 bronze badges

$\\endgroup$

0

include a comment |

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

Featured top top Meta

Linked

20

Is the null collection a subset the every set?

0

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

0

Set theory; sets and subsets; Is one empty collection contained within a collection that includes real numbers?

0

Any collection A has actually void collection as its subset? if correct how?

related

10

straight proof the empty collection being subset of every set

3

If the empty collection is a subset the every set, why isn't $\\\\emptyset,\\a\\\\=\\\\a\\\\$?

1

A power set contais a set of a north subset?

3

How have the right to it be the the empty set is a subset the every set but not an element of every set?

3

Is the set that includes the empty set ∅ likewise a subset of every sets?

hot Network concerns much more hot concerns

tastecraftedmcd.comematics

company

stack Exchange Network

site design / logo design © 2021 stack Exchange Inc; user contributions licensed under cc by-sa. Rev2021.9.27.40308

tastecraftedmcd.comematics stack Exchange works finest with JavaScript allowed

your privacy

By click “Accept every cookies”, friend agree stack Exchange deserve to store cookie on your maker and disclose info in accordance v our Cookie Policy.