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Let us consider the following 3 propositions

1) ($n$ is good) $\iff (x=y)$

2) ($n$ is good) $\iff (x=z)$

3) ($n$ is good) $\iff (x=y=z)$

It is clear that from 1,2 we can obtain 3.

My doubt is whether is it possible to obtain 1 from 2,3 or similarly 2 from 1,3?

$\iff$ stands for if and only if

= stands for standard equal to

Thanks in advance :)

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  • $\begingroup$ Put some quantifiers in there, this makes no sense. $\endgroup$ – Git Gud Dec 31 '13 at 18:08
  • $\begingroup$ @GitGud Is it okay now ? $\endgroup$ – hanugm Dec 31 '13 at 18:12
  • $\begingroup$ Not really, what about quantification on $x,y,z$? And for all $n$ where? In $\mathbb N$? $\endgroup$ – Git Gud Dec 31 '13 at 18:16
  • $\begingroup$ @GitGud ofcourse, But instead of quantification just treat them as propositions or else universal quanitifier for all 3 propositions. $\endgroup$ – hanugm Dec 31 '13 at 18:20
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    $\begingroup$ How did this get $5$ up votes? $\endgroup$ – Git Gud Dec 31 '13 at 21:06
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To simplify, let's replace your propositions (which we have no context for) with propositional variables.

$$G: n\text{ is good}$$ $$P: x = y$$ $$Q: x = z$$

Note that $x = y = z$ is equivalent to $P\land Q$.

Now we have

(1) $G\leftrightarrow P$

(2) $G\leftrightarrow Q$

(3) $G\leftrightarrow P\land Q$

We cannot conclude (1) from (2) and (3): Consider the case when $P$ is true but $G$ and $Q$ are false.

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