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I've come across an exercise which requires you to express every one of the logical operators $\{\wedge, \Rightarrow, \Leftrightarrow\}$ with the other two - for example, $$p \Leftrightarrow q \sim (p \Rightarrow q) \wedge (q \Rightarrow p).$$ So far I've only been able to do this with $\Leftrightarrow$, which is straightforward enough, but now I'm stuck. I don't think I'm allowed to use constants such as '0' or '1'. Is this even possible? Any help (tips or a solution) would be greatly appreciated.

Edit: ok I found the solution for $\wedge$:

$$p \wedge q \sim (p \Leftrightarrow q)\Leftrightarrow ((p \Leftrightarrow q)\Rightarrow p)$$

the_dude
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1 Answers1

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I think that works:

$$p\Rightarrow q \sim p\wedge q \Leftrightarrow p\\ p\wedge q\sim (p\Rightarrow q)\Leftrightarrow (q\Rightarrow p)$$

(You can check truth tables)

NHL
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    The first one seems correct, but the second one doesn't (for example, if $p=q=0$). Still, thanks for your answer. – the_dude Jan 29 '21 at 16:23