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Im having quite a bit of trouble understanding the Domination and Contraposition Laws in the instance below. I just do not see how the Domination Law, $\rho \wedge \mathrm{F} \leftrightarrow \mathrm{F}$ or $\rho \vee \mathrm{T} \leftrightarrow \mathrm{T}$, works at all on line 5. I also am confused on how the Law of Contraposition is working in line 6. Please help me understand this better. Thank you!
How is this domination

woody
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    Would you be happier with $\sim s \to ;(\sim s) \vee (\sim r) \to ; \sim (s \wedge r)$ ? – Henry Aug 05 '14 at 21:47
  • I would if i understood that transformation! – woody Aug 05 '14 at 21:51
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    In my version you go from something true $(\sim s)$ to its union with something else and thus truer by domination. Then rewrite with De Morgan. – Henry Aug 05 '14 at 21:53
  • I can see what you mean however, you used a $\vee$ which is an or statement. You can only combine lines, as I understand, with a $\wedge$ and statement. – woody Aug 05 '14 at 22:03

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Line 3 states $\sim\!s$, i.e. that $s$ is false. Then use domination rule with $\rho=r$, and regard $s$ as $F$: $$s\land r \equiv F$$ which is just the 5th line [$\sim\!(s\land r)$], and it came only from line 3.

The 6th line is derived from the 4th and 5th line by the law of Contraposition: if something of the form $A\to B$ is proven and $\sim\!B$ is also known, then $\sim\!A$ follows. Take $A=(\sim\!p\,\lor\sim\!q)$ and $B=(s\land r)$.

Berci
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  • Ah Ha! I was thinking along the lines of if something is negated it is referred to as false. I'm still a little confused on the transition of statements. The way I am thinking of this is more like addition, ie. line 5 refers to adding of line 3 and 4 using a $\wedge$ on the left side inbetween them. I just don't understand what happened to the left side? The second part makes complete sense although I would think the proper line comment would be Contraposition (4,5)? – woody Aug 05 '14 at 22:59
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    Upon looking at this with a fresh mind it made perfect sense thank Berci! – woody Aug 06 '14 at 14:05