Question

why empty set is a subset of every set

Answer 1

think of the empty set as being equivalent to 0 in 'ordinary' algebra or arithmetic
adding 0 to x, y or 5 for example doesnt change their value
an empty set can be part of any set and set does not change.

Answer 2

Given a set X, another set Y is defined to be a subset of X if
\[x \in Y \implies x \in X\]

For example, is X = {1, 2, 4, 8} and Y = {2, 4}
Y is a subset of X because
\[2 \in Y \ \ and \ \ 2 \in X\]
\[ 4 \in Y \ \ and \ \ 4 \in X \]
and hence it is true that
\[x \in Y \implies x \in X\]
But the set Z = {2, 4, 6} is not a subset of X because it is not true that
\[x \in Z \implies x \in X\]
Namely
\[6 \in Z \ \ but \ \ 6 \notin X\]

Now consider the empty set. Is it true that
\[x \in \emptyset \implies x \in X\]
In a trivial sense this is true because there are no x in the empty set at all so there is nothing to check.

This might feel like some sort of trick of logic but actually it is makes complete sense and we can see this another way.

Given our set X with three members--2, 4, and 8--a subset of X will each member of X or not.

So one obvious subset of X is the set will all member of X: {2,4,8}
Now there's a subset without 8: {2,4}
or another subset without 2 and 8: {4}
or another subset with 8 but not 2 and 4: {8}
There's also a subset of X without any of 2, 4 and 8: {}.
This last subset is the empty set. Hence it is perfectly consistent with what a subset is to have a subset which has none of the members.

Ok, I'll shut up now.