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please help I'm stuck. I'm trying to solve this.

enter image description here

so far I have:
a) a+b+c
b) a+bc
c) a+b
d) a+b

but for e) I can't progress further since I don't know how to deal with a'bc in this case. anyone so kind to please help me?

Shads
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2 Answers2

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$$a + \overline abc = a + bc$$ since we have $a$ or $\overline a bc$. If not $a$, then $\overline a$. That must follow if not a. So "it goes without saying", that if not $a$, (then we already know $\overline a$) so it suffices to assert $bc$.

A similar argument can be made for $bc + \overline bc$, but perhaps simpler, in this case, we can use the distributive property: $$bc + \overline b c = (b +\overline b)c = 1c = c$$

Combining the simplifications gives us $$a+\bar a b c+\bar b c=a+c$$

amWhy
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  • that makes sense finally! may I ask if you can see if the correct answer for f) is ab+a'c? – Shads May 03 '14 at 13:00
  • I argue with my classmate. he says the answer for f) is ab+c? – Shads May 03 '14 at 13:16
  • The correct answer is $ab + c$. I just double checked, and arrived at $ab+ c$. – amWhy May 03 '14 at 13:30
  • I made it until ab+(a'b')*c. how do you get rid of the a'b' please? – Shads May 03 '14 at 13:32
  • $ab + a'c + b'c = ab + (a'+b')c$. Now, if $ab$ is false, then $\lnot (ab) = a' + b'$ is true, meaning either $a'$ is true, or b' is true. In either case, $a' + b'$ is true. So either $ab$ is true, or if false, then we already know $a'+b'$ must be true, in which case it suffices to declare that $(a'+b')c$ is true provided $c$ is true. If $ab$ is false, the truth value of $(a'+b')c$ depends only on $c$. – amWhy May 03 '14 at 13:36
  • omg now that makes sense....thank you so much. for g) it's a+c'? – Shads May 03 '14 at 13:47
  • It should be $a + b' +c'$ – amWhy May 03 '14 at 16:04
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$a+\bar a b c= a+bc$ and $bc+\bar b c=c$, so $a+\bar a b c+\bar b c=a+c$

Hagen von Eitzen
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  • thank you very much for the quick answer. but I can't quite follow how you get from a+a'bc=a+bc. a+a'= 1 if you pull A out it would be for me a(1+bc)? – Shads May 03 '14 at 12:24