Proving $A+A'B=A+B$ without truth tables

Question

How can I prove the Boolean algebraic rule $$A+A'B=A+B$$ without using a truth table?

With the truth table, it is easy to see that the two are equal, but how can I prove it using lesser Boolean identities?

Answer 1

Use the distributivity of $\lor$ (sum) over $\land$ (product), which is unique to Boolean algebra: $$A+A'B=(A+A')(A+B)=A+B$$