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How can I use mathematical notation to say that an equation doesn't need to be true? For example, take the set $A = \{ (a_1, a_2, a_3, a_4) \mid a_1a_2 = 0 \}$. $A$ isn't a vector subspace of $\mathbb{R}^4$, because $u,w \in A$ does not imply $u+w \in A$. Indeed $(u_1+w_1)(u_2+w_2)$ isn't always equal to zero always, but sometimes it is, so I can't say that $(u_1+w_1)(u_2+w_2) \neq 0$. Is there any way to express this with mathematical notation?

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  • $\begingroup$ Usually one says something on the line “generally $(u_1+w_1)(u_2+w_2)=0$ doesn't hold”. There's no need for a specific notation. $\endgroup$ – egreg Sep 30 '17 at 10:22
  • $\begingroup$ If what you want to do is to prove that $A$ is not a vector subspace, the best thing is to give a concrete example, i.e. actual numbers for which the equation does not hold. $\endgroup$ – Eike Schulte Oct 1 '17 at 17:45
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For example, $$f(x)=\begin{cases} 1 &\text{ if } x\in \mathbb{Q}\\0 &\text{ if } x\notin \mathbb{Q}.\end{cases}$$

Modify to suit your needs, in any possible way.

For example, $$A \in {\mathbb M_{n \times n }}=\begin{cases} 1 &\text{ if } \det(A) >0 \\0 &\text{ if } \det(A)=0\\ -1&\text{ if } \det(A)<0 \end{cases}$$

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