TITLE: In boolean algebra, why is a+a'b = a+b?
QUESTION [8 upvotes]: Looking at the truth table, they're equivalent:
A   B      (A+B)  (A+A'B)
-------------------------------
1   1       1       1
1   0       1       1
0   1       1       1
0   0       0       0

But what manipulation can one do using basic identifies and laws to show that they're the same?

REPLY [17 votes]: Note with the laws of Boolean algebra, "addition" distributes over "multiplication" (just as multiplication would normally distribute over addition).  Thus, we have
$$
a + (a'\cdot b) = (a+a')\cdot (a+b) = 1(a+b) = a+b
$$