boolean-logicboolean-expressionboolean-algebra

How the expression !A + (A . !B) = !(A.B)?

I have an expression !A+(A.!B) and on an expression solver, it gives the !A+(A.!B) = !(A.B)?. The solver notified that "Apply the Absorption Law" A.B+!A = B+!A.

I have made the truth tables for both the expressions and the answer was correct. But the problem is I can not understand how the absorption law has got implemented to my expression !A+(A.!B)?

Can someone please explain in details how the absorption law has got implemented to my expression?

Solution

  • I am going to assume that + = OR, . = AND and ! = NOT.

    The absorption law was applied in the very first step:

      !A + A.!B
= !A + !B     (if the first monomial does not hold, A is "true"
               and thus does not need to be checked again)
= !(A.B)      (De Morgan's rule)