> The subset relation is defined as "X is a subset of Y if for all x in X, x is in Y".
This might be a bit nit picky, but shouldn't that be "if and only if" or "iff" instead of just "if"?
> The negation of this is "X is not a subset of Y if there exists some x in X, such that x is not in Y". We can see from this that the empty set is NOT NOT a subset of every set.
In what follows I'm replacing lower case x with lower case e.
I'm kind of simple minded, so find that a bit confusing because of using the negated definition and a double not, and also "for all e in X" when X is the empty set because it makes you have to keep for the rest of the definition that e might be nonexistent.
This is clearer to me:
1. Rewrite the definition so it is in terms of all e, not all e in X: "X is a subset of Y if and only if for all e, e is in X implies e is in Y". This form of the definition only talks about elements that exist.
2. "e is in X implies e is in Y" is equivalent to "e is not in Y implies e is not in X", which gives us this equivalent definition: "X is a subset of Y if and only if for all e, e is not in Y implies e is not in X".
3. When X is the empty set it is always the case that e is not in X, and so "e is not in Y implies e is not in X" is always true (because both false implies true and true implies true are true), and so X is a subset of Y by the definition of subset.
This might be a bit nit picky, but shouldn't that be "if and only if" or "iff" instead of just "if"?
> The negation of this is "X is not a subset of Y if there exists some x in X, such that x is not in Y". We can see from this that the empty set is NOT NOT a subset of every set.
In what follows I'm replacing lower case x with lower case e.
I'm kind of simple minded, so find that a bit confusing because of using the negated definition and a double not, and also "for all e in X" when X is the empty set because it makes you have to keep for the rest of the definition that e might be nonexistent.
This is clearer to me:
1. Rewrite the definition so it is in terms of all e, not all e in X: "X is a subset of Y if and only if for all e, e is in X implies e is in Y". This form of the definition only talks about elements that exist.
2. "e is in X implies e is in Y" is equivalent to "e is not in Y implies e is not in X", which gives us this equivalent definition: "X is a subset of Y if and only if for all e, e is not in Y implies e is not in X".
3. When X is the empty set it is always the case that e is not in X, and so "e is not in Y implies e is not in X" is always true (because both false implies true and true implies true are true), and so X is a subset of Y by the definition of subset.