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Let $E, F$ be two sets.

1) If $E$ is empty and $F$ is nonempty, is their difference $E \setminus F$ meaningful?

2) If $E, F$ are both nonempty and disjoint, is their difference $E \setminus F$ meaningful?

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3 Answers 3

up vote 4 down vote accepted

Yes to both questions.

Recall that $E\setminus F=\{x\in E\mid x\notin F\}$. In both cases the answer is fairly easy to calculate, and I'll leave it for you to do so.

The point is that there is meaning as long as $E$ and $F$ are sets. And the empty set is a set (proof by terminology).

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Oh much appreciated, I get it. – Gudson Chou Jul 24 '14 at 4:35
"proof by terminology" is a rare phrase to be used carefully. A spherical triangle does not have all the properties usually assumed for triangles. – Henry Jul 24 '14 at 7:20
@Henry: That's a joke really. Here's a nice anecdote about this sort of proofs. I took a course in non-commutative algebra, and when we reached the Brauer Group it was first called "Brauer Set" until we proved it is in fact a group. Why? Because two (or more?) years before, when the professor gave the course, someone jokingly said that we don't have to prove it's a group because it is called a group, so it must be a group. :-P – Asaf Karagila Jul 24 '14 at 7:52

Hint: If you think of forming $E\setminus F$ as "start with everything that's in $E$, then remove everything that's also in $F$", you should be able to answer both of these questions easily.

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To aid the above answers with some intuition, think of $E \backslash F$ as the set $E$ "without" elements of $F$, i.e. $E \cap F^c$, the set of all elements of $E$ that are not in $F$.

So, as for the first, let $E$ be the set of United States presidents prior to the current date that were female. Then $E$ is (shamefully) empty. Let $F$ be the set of all red-headed humans, past and present. What, then, are the elements in $E \backslash F$, the set of female presidents who were not red-headed?

As for the second, let $E$ be the set of all even numbers, and $F$ be the set of all odd numbers. Then, what set is $E \backslash F$, the set of all even numbers that are not odd numbers?

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