Take for instance the set $X=\\a, b\\$. Ns don\"t watch $\\emptyset$ anywhere in $X$, for this reason how deserve to it be a subset? $\\begingroup$ \"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

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edited Jun 25 \"19 in ~ 13:51
answered january 29 \"14 at 21:55 Mauro ALLEGRANZAMauro ALLEGRANZA
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$\\begingroup$
Subsets room not necessarily elements. The elements of $\\a,b\\$ space $a$ and $b$. Yet $\\in$ and $\\subseteq$ are various things.

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answered january 29 \"14 at 19:04 Asaf Karagila♦Asaf Karagila
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