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if $(\forall\ x \in \mathbb{R})(x=x) \rightarrow\ (\forall\ x\in \mathbb{R})(x/x = 1) $

But this is not true because $x$ cannot equal zero for the second statement. Why?

I understand that we cannot divide by zero, but aren't these statements logically equivalent?

Gary
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KOR
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  • They are not logically equivalent. Only the statement $\forall x \in (\mathbb{R} \setminus 0), \frac{x}{x}=1$ is true. – Scene Nov 09 '22 at 07:23
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    No. They are not equivalent. Why precisely do you think they are? $\frac xx = 1$ implies two things that must be true. a)$x = x$ and b) $x\ne 0$. And $x=x$ implies one or the other of two things which only one or the other can be true: EITHER $\frac xx =1$ OR $x = 0$. Basically $(\forall x\in \mathbb R(x=x)\iff (\forall x\in \mathbb R) \frac xx =1$ OR $x=0$ and that is a true statement. While $(\forall x\in \mathbb R)(\frac xx=1)\iff (\forall x\in \mathbb R)x=1$ AND ($x\ne 0$ OR $\frac 00=1)$. That is simply false. The two statements are not the same and are not equivalent. – fleablood Nov 09 '22 at 07:37

1 Answers1

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They are not equivalent.

The truth values for $x=x$ and $\frac xx =1$ are dependent upon whether $x = 0$ or not and the truth tables are :

$\begin{array}\ x=0 &|&x=x&|&\frac xx =1\\T&|&T&|&F\\F&|&T&|&T \end{array}$

So that simply are not equivalent. They just aren't.

And $(\forall x\in \mathbb R)x=x$ is simply true. and $(\forall x\in \mathbb R)\frac xx=1$ is simply false.

....

However the statements $(\forall x\in \mathbb R\setminus\{0\})x=x$ and $(\forall x \in \mathbb R\setminus\{0\})\frac xx=1$ are equivalent.

That truth table has simply one line:

$\begin{array}\ x\in \mathbb R\setminus\{0\}&|&x=x&|&\frac xx = 1\\T&|&T&|&T\end{array}$

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Also the statements $(\forall x\in\mathbb R)x=x$ and $(\forall x\in \mathbb R)\frac xx=1$ OR $x=0$ are equivalent (and both true).

And the statements $(\forall x\in \mathbb R)\frac xx=1$ and $(\forall x\in \mathbb R)[x=x$ AND $x\ne 0]$ are equivalent (but both false).

fleablood
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