$\\begingroup$ \"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
edited Jun 25 \"19 in ~ 13:51
answered january 29 \"14 at 21:55
Mauro ALLEGRANZAMauro ALLEGRANZA
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Subsets room not necessarily elements. The elements of $\\a,b\\$ space $a$ and $b$. Yet $\\in$ and $\\subseteq$ are various things.
answered january 29 \"14 at 19:04
Asaf Karagila♦Asaf Karagila
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