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Consider following cases:

  • example #1

$$\forall x,y\in\mathbb R\left(\neg\left(\frac{1}{x}\ge y\right)\rightarrow\frac{1}{x}\lt y\right)$$

  • example #2 (assuming universe of discourse is all square matrices elements of which are real numbers) $$\forall A,B\left(B=B (A A^{-1})\right)$$ As for example 1 if x is zero then what does the statement evaluate to and as for example 2 what if A and B are not conformable for multiplication or A does not have an inverse? Are these considered undefined or false?
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    To evaluate definiteness of statement without providing the language is indecent and it makes this an ill-posed problem. I, for instance, have never seen a language which includes $x^{-1}, y^{-1}, \ldots$, nor $\frac 1 x$, etc. I'd say they are both undefined. **Edit:** Since the universe is that of square matrices, the product always makes sense. – 2017-02-19
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    @GitGud Well it says all square matrices not square matrices of a specific size so we encounter multiplication of a 5*5 matrix by a 8*8 matrix as an example which is undefined – 2017-02-19
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    Of course, you're right. Missed that. – 2017-02-19
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    Even if we limit universe of discourse to a set of square matrices of a specific size the issue of A not having inverse still persists – 2017-02-19
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    For the first case, what with $x=0$? – 2017-02-20

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