I have these expressions
$$(A'+B'+C'+D')(A'+B'+C'+D)(A'+B+C'+D')(A'+B+C'+D)(A'+B+C+D')(A+B'+C'+D')(A+B'+C'+D)(A+B+C'+D')(A+B+C'+D)$$ $$(A'B'C D')+(A'B'C D)+(A'B C D)+(A B'C D')+(A B'C D)+(A B C D)'+(A B C D)$$
And need to simplify both. Addition means $\lor$, multiplication means $\land$ and the prime means $\neg$.
From a similar question that I asked before I know that I have to follow the laws of Boolean algebra, But I do not know how to apply them in this case that I have four variables. I think i can do this in wolframalpha but i really want to know how to do it
The first expression reduces to the Boolean product (logical AND) of these five factors $$(A'+B'+C'+D')(A'+B'+C'+D) = A'+B'+C'$$ $$(A'+B+C'+D')(A'+B+C'+D) = A'+ B + C'$$ $$(A+B+C'+D')(A+B+C'+D) = A + B + C' $$ $$(A+B'+C'+D')(A+B'+C'+D) = A + B' + C' $$ $$ (A'+B+C+D') $$ Looking at the first four, we see that the Boolean product (logical AND) of all four must be $C'$ since all four factors must evaluate to true, and the only way this is possible is if $C'$ is true. So the expression reduces to: $$ C'(A'+B+C+D') = C'(A'+B+D') $$
The second expression is the logical OR of multiple terms, but looking at the last two: $$(A B C D)' + (A B C D) = 1$$ So the whole logical OR reduces to $1$ (or logical TRUE).