help with boolean algebra

Everything In Its Right Place

Im having trouble breaking down this equation. Could someone help me out?

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F = AD(D + AC) + C(A+C + C)

Everything In Its Right Place

sorry the over score is actually over the A+C in the second set of brackets

japester

F = AD(D+AC) + C(!(A+C) + C)


Distributive law => X(Y+Z) = XY + XZ therefore
AD(D+AC) = ADD + ADAC


XX = X always therefore DD = D and AA = A so


ADD + ADAC becomes AD + ADC (left hand side dealt with for now)


!(A+C) = !A!C by DeMorgan’s Theorem and so, by the distributive law, the expression
C(!(A+C) + C) becomes C!A!C + CC


C!C is always 0 and CC is always C so it simplifies to


A!0 + C but X0 is always 0 so it simplifies further to just


C (right hand side dealt with for now)

Combining both results we have:


AD + ADC + C


Using the distributive law we can write it as:


AD(1 + C) + C


But 1+C is always 1 so we have the original expression simplified to just


AD + C


hope this helps, Japester

Everything In Its Right Place

thanks!

First Explorer

Ah the joys of boolean algebra, was a major stumbling block for a lot of people in first year college!