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boolean-logicboolean-algebra

What is the simplified boolean expression for A+A`B`?

I understand that

A+A`B=A+B

But what happens when

B

is replaced by

B`

on LHS?

Solution

  • The reason A + A'B is equivalent to A + B is because the A' is superfluous given the A + preceding it. If A' weren't already true then A would be so the A + would already make everything true. Anything could follow A' in the second part and the answer would be A + the rest. In your case, A + A'B' is equivalent to A + B'. We can verify with a truth table:

    A    B    A'    B'     A'B'    A + A'B'    A + B'
T    T    F     F      F       T           T
T    F    F     T      F       T           T
F    T    T     F      F       F           F
F    F    T     T      T       T           T

    Because A + A'B' and A + B' have the same values for all possible inputs, they are equivalent.