Solving Boolean Algebra: a'b'c' + abc = 1?

[killerfish]

Hi guys,

I'm new to boolean algebra, i couldn't get this through...

a'b'c' + abc = 1 ? or i have to use (abc)' + abc = 1 to get 1 ?

Thanks you.

 

[Shackman]

(abc)' + abc = 1. There are only 2 possible values for abc, and they are 0 and 1. So it is very obvious that if abc isn't 1, then (abc)' is, and vice versa. So 1 or 0 = 1, 0 or 1 = 1. To prove that a'b'c' + abc = 1 isn't always necessarily true, you can construct a truth table with a b c a' b' c' and your answer. You will find that statement is only true when a,b,c are all the same value.

 

[Mene]

Hello there,

Boolean algebra is a type of algebra that deals with logical expressions using the values of true and false, represented by 1 and 0 respectively. In order to solve this expression, we need to understand the basic rules of boolean algebra.

Firstly, the expression a'b'c' + abc can be simplified using the distributive law, which states that a(b+c) = ab + ac. In this case, we can distribute the a' over the two terms in the parentheses, giving us a'b'c' + a'bc. Similarly, we can distribute the b' over the two terms in the parentheses, giving us a'b'c' + a'b'c. Finally, we can distribute the c' over the two terms in the parentheses, giving us a'b'c' + a'b'c + a'bc.

Now, we can see that there are two terms that are the same (a'b'c'), and according to the associative law, we can rearrange the terms without changing the value. Therefore, we can rewrite the expression as a'b'c' + a'b'c + a'bc = a'b'c + a'bc.

Next, we can use the identity law, which states that a + 0 = a, to simplify the expression further. In this case, we can add a'b'c to the expression, giving us a'b'c + a'b'c + a'bc = a'b'c + a'b'c + a'bc + a'b'c. Now, we can see that there are three terms that are the same (a'b'c), and we can use the distributive law again to rewrite the expression as a'b'c + a'bc + a'b'c = a'b'c + a'bc.

Finally, we can use the complement law, which states that a + a' = 1, to simplify the expression even further. In this case, we can use the complement of a'b'c, which is abc, giving us a'b'c + a'bc = abc + a'bc.

Now, we have only two terms left, and we can use the distributive law one more time to combine them into one term, giving us abc + a'bc = (a+a')bc = 1bc = bc.

Therefore, the simplified expression is bc = 1, or