Analyze if the following statement is true or false: Suppose $(D,+,\cdot,',0,1)$ is a Boolean Algebra. Then $a'b+(abc)'+c(b'+a)=b'+c'$.
My guess is that the statement is false.
Let $D=D_{15}$ be the divisors of $15$. The Hasse diagram is
We know that $(D,+,\cdot,',0,1)$ is a Boolean Algebra since, by a property, $15$ can be expressed as a product of unique primes. So let for example $a=1$, $b=3$ and $c=15$. Then $a'=15$, $b'=5$ and $c'=1$, thus \begin{align*}a'b+(abc)'+c(b'+a)&=15\cdot3+(1\cdot3\cdot15)'+15\cdot(5+1)\\&=3+15+5\\&=15,\end{align*} but \begin{align*}b'+c'&=5+1\\&=5,\end{align*} hence the proposition is false.
Another way that occurred to me to prove that the proposition is false is through the truth tables of each expression, knowing that $\cdot$ represents the $\mathrm{AND}$ logical and $+$ is the $\mathrm{OR}$ logical. So: $$\begin{array}{|c|c|c|c|c|c|c|c|c|c|c|} \hline a&b&c&a'&b'&c'&a'b&abc&(abc)'&b'+a&c(b'+a)&a'b+(abc)'+c(b'+a)\\\hline \mathrm T&\mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm F&\mathrm F&\mathrm T&\mathrm F&\mathrm T&\mathrm T&\mathrm T\\\hline \mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm F&\mathrm T&\mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm F&\mathrm T\\\hline \mathrm T&\mathrm F&\mathrm T&\mathrm F&\mathrm T&\mathrm F&\mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm T&\mathrm T\\\hline \mathrm T&\mathrm F&\mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm F&\mathrm T\\\hline \mathrm F&\mathrm T&\mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm T&\mathrm F&\mathrm T&\mathrm F&\mathrm F&\mathrm T\\\hline \mathrm F&\mathrm T&\mathrm F&\mathrm T&\mathrm F&\mathrm T&\mathrm T&\mathrm F&\mathrm T&\mathrm F&\mathrm F&\mathrm T \\\hline \mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm T&\mathrm T \\\hline \mathrm F&\mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm F&\mathrm T \\\hline \end{array}$$ but $$\begin{array}{|c|c|c|c|c|} \hline b&c&b'&c'&b'+c' \\\hline \mathrm T&\mathrm T&\mathrm F&\mathrm F&\mathrm F \\\hline \mathrm T&\mathrm F&\mathrm F&\mathrm T&\mathrm T \\\hline \mathrm F&\mathrm T&\mathrm T&\mathrm F&\mathrm T \\\hline \mathrm F&\mathrm F&\mathrm T&\mathrm T&\mathrm T \\\hline \end{array}$$ As we can see, both truth tables do not produce the same truth values, hence the proposition is false.
Are both options correct to prove that each one separately proves the falsehood of the proposition?
Thanks!!
The statement is false. Besides the methods you used, there are various other.
Karnaugh map is full of 1s.
BA rules applied to the LHS give $$\begin{aligned} a'b+(abc)'+c(b'+a)&=a'b+ a'+b'+c'+cb'+ca\\&=(a'b+ a')+(b'+cb')+c'+ac\\ &=a'+b'+c'+ac\\ &=(a'+a)(a'+c)+b'+c'\\ &=a'+c+b'+c'=1\end{aligned}$$ which is clearly not equal to RHS.