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I'd like to find out the equivalence existing between these two logical functions, I was trying it just by applying Boole's theorems in just one of them, but still have not arrived to a conclusion yet. $$\overline{ab}h + \overline{c}fgh + \overline{d}fgh + \overline{e}fgh = \overline{(a+b)(cde+ \overline{fg})}h$$

How should Boole's theorems be applied to demonstrate the equivalence between both functions?

Thanks so much in advance.

Astyx
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edusola93
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  • I think you should evaluate the right-hand side to see if you get the left-hand one. – Wuestenfux May 21 '17 at 06:52
  • well @Wuestenfux thanks, already tried that but somehow can't arrive to the solution which would be getting the same on both sides, i'll keep waiting if someone can find out and help me :) – edusola93 May 21 '17 at 10:07

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If you don't mind, I'll use accent marks instead of overline's ...

$$[(a+b)(cde+(fg)')]'h=\text{ (DeMorgan)}$$

$$[(a+b)'+ (cde+(fg)')']h= \text{ (DeMorgan)}$$

$$[a'b'+((cde)'fg)]h = \text{ (DeMorgan)}$$

$$[a'b'+(c'+d'+e')fg]h = \text{ (Distribution)}$$

$$a'b'h+(c'+d'+e')fgh= \text{ (Distribution)}$$

$$a'b'h + c'fgh+ d'fgh+ e'fgh$$

OK, so that is almost, but not quite the same as your left side, which seems to be:

$$(ab)'h +c'fgh+d'fgh+e'fgh$$

that is: that first term is a little different. But: with these overlines it is sometimes hard to tell the difference between $\overline{ab}$ and $\overline{a}\overline{b}$ (which is exactly why I personally avoid overlines)... So I think that is what is going on!

Bram28
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  • Damn, I was mistaken at two accent marks while I was trying to do it, this is just perfect man, thanks so much for the answer! But, wouldn't the 4th line be DeMorgan too when you go from cde to c'+d'+e'? or is that really Distribution? Thanks @Bram28 ! – edusola93 May 21 '17 at 17:26
  • @edusola93 Sorry, I have a peculiar notation where I put the justification between the sentences, so from 1 to 2, 2 to 3, and 3 to 4 are all DeMorgan's, and only 4 to 5 and 5 to 6 are Distribution. So yes, you're right, going from 4 to 5 is Distribution. – Bram28 May 21 '17 at 23:43
  • no problem! Thanks a lot! – edusola93 May 23 '17 at 16:22