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## X = \left\{ (x_1, x_2, x_3) \in \mathbb{F^3}:x_1 x_2 x_3 = 0 \right\} ##

I know that I can simply provide a counterexample and show that the subset X above is not closed under addition -- namely, I can construct two vectors who are elements of X such that their sum is not an element of X (for instance, I could take the vectors (2, 0, 0) and (0, 3, 3) in X, whose sum (2, 3, 3) is clearly not in X. While I know such an argument would be entirely legitimate, I wondered if there were a more general way of demonstrating whether the subset is a subspace of ##F^3##.

For instance, could I construct an argument centered on the nonlinearity of the function ##f(x_1, x_2, x_3) = x_1 x_2 x_3 ##, showing that in general ##f(x + y) \neq f(x) + f(y) ##?