When an equation has no solutions, denote it with $x\in\varnothing$. My teacher claims that when an equation in variables $x_1,x_2,\ldots,x_n$ has no solutions, you should denote this fact with $(x_1,x_2,\ldots,x_n)\in\varnothing$.
An empty set can't have an element in it, so this can't be right.
 A: You can form the set $L$ of all solutions. So iff $(x_1, \cdots, x_n)$ is a solution, then $(x_1, \cdots, x_n)\in L$. Now, if there are no solutions, then $L = \varnothing$. Insert that, and you get $(x_1, \cdots, x_n)\in \varnothing$.
In other words, the sequence $(x_1, \cdots, x_n)$ is a solution to an impossible set of equations iff $(x_1, \cdots, x_n)\in \varnothing$.
However, saying that "$(x_1, \cdots, x_n)\in \varnothing$ means there are no solutions" is taking it a bit far. I would rather say "$L = \varnothing$ means there are no solutions".
A: You are correct that the empty set is the set with no members. The solution set of an equation is, however defined as the set of all solutions that satisfy the equation and since that set has no members the solution set is empty.
Thus, saying that the equation has no solution, and saying that the solution set of the equation is empty mean the same thing.
Recall that the solution set is the set of all values that turn an open sentence into a true statement.
Thus, the membership symbol is an abbreviation for the fact that the variable can be replaced by a value that is "in", "belongs to", or "is a member of" the set. If the set is empty, then the variable can not be replaced by any value. This is what it means for an equation to have "no solution", there is no value that when substituted for the variable will result in a true statement.
I agree with you that the notation does lead to some confusion when dealing with the special case of the solution set being empty. In this case "what the variable can be replaced by belongs to" a set with no members. But, if you like, you can simply say "No solution." Note that the meaning here is still the value that the variable can be replaced by belongs to the replacement set, which happens to be empty so there is no value.

variable: A symbol used to represent any member of a given set.


domain of a variable: The set whose members may serve as replacements for the variable; also called replacement set.

A: I would write
$card\{X
\mid f(X) = 0\}
= 0
$
where
$card\{S\}$
is the number of elements in a set $S$
and
$X$ is a vector
$(x_1, x_2, ..., x_n)$.
Another way would be
$\{X
\mid f(X) = 0\}
=\varnothing
$.
A: A more efficient way to denote that the equation has no solution, is by a Boolean "false", "F", or "$\bot$", consistent with your observation that the empty set cannot have any elements.
