Solving a boolean expression

Question

I am trying to solve the following Boolean expression:

$$a + \neg{a} b + \neg{a} \neg b c + \neg a \neg b \neg c d + \dots$$

The question asked was to use Boolean algebra in order to solve the above expression.

My attempt so far:

$$y = a + \neg a b + \neg a \neg b c + \neg a \neg b \neg c d = a + \neg a y$$

This does not look meaningful to me.

Answer 1

First show that p + ¬pq =p + q. Then a + ¬ab = a + b by letting p= a and q=b.

Now let p = a+b and q = c to get a + ¬ab + ¬a¬bc = a + b + c since ¬p = ¬a¬b by De Morgan. Continue inductively.

Answer 2

I assume this is an infinite Boolean expression. Regardless of that, some natural-language reasoning tells us it should simplify to $a + b + c + d + \dots$.

("This dog is either a black lab, or not a black lab and a golden retriever, or not a black lab and not a golden retriever but a Pomeranian, or ..." is a verbose way of saying that you know nothing about this dog.)

Answer 3

Writing $a_k$ instead of $a,b,c,\dots$ and shorter $\bar a$ instead of $\neg a$, we have $y_n=a_n+\bar a_ny_{n+1}$. Plugging in values for $a_n$ we get $y_n=a_n+y_{n+1}$. So for fixed $N$ we have $y_0=\sum_{n=0}^{N-1}a_n+y_N$.