In Linear Algebra Done Right by Sheldon Axler one of the exercises is as follows:
"The empty set is not a vector space. The empty set fails to satisfy only one of the requirements listed in 1.19. Which one?"
1.19 lists the requirements of a vector space. I.e. associativity, commutativity, additive identity, additive inverse, multiplicative inverse, and distributive property.
The answer to the question is that the empty set doesn't satisfy additive identity due to the fact that there is no element $\mathbf{0}$ in vector space $\mathbf{V}$.
How does one going about proving that the empty set meets the other requirements of associativity, commutativity, additive inverse, multiplicative inverse, and the distributive property?
I'm confused about how the elements of an empty set can be shown to follow these properties when they don't exist.
Best Answer
To add to the other answers, one way I like to look at this this situation is as follows:
All the of the axioms you list: associativity, commutativity, additive inverse, etc... all start with the quantifier "for all". In other words, if you wrote them out with heavy notation they would have the form $\forall x P(x)$ where $P(x)$ is some further property of $x$. To say that a space fails to satisfy such a statement amounts to saying that the space satisfies the negation of the statement, which has the form $\exists x Q(x)$ where $Q(x)$ is the negation of $P(x)$. So right away we see that an empty space cannot satisfy such a statement since it's asserting that there is some element of the space.