If X is a subset of Y then all elements of X should be present in Y.

i.e. X ∩ Y = X

Let us take X = Φ, then by definition, X does not contain any element.

Thus the all the common elements between X and Y, in this case, are none.

In other words, the set of common elements between X and Y is the null set or X.

Thus, X ∩ Y = X is satisfied.

Or Φ is the subset of Y.

This is true even if Y is a null set.

Ex: Let M be the set of all dogs that know Physics.

Now, M should be a null set.

Again, Is this set a subset of the set of all scientists?

The answer is Yes because there are no such dogs.

So, Φ is a subset of every set.