In boolean algebra, why is a+a'b = a+b? [duplicate]

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Looking at the truth table, they're equivalent:

A B (A+B) (A+A'B)
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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?

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2 Answers

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$A+A'B=A(1+B)+A'B=A+AB+A'B=A+(A+A')B=A+B$

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

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