Not new to electronics, but new to forum. Coming back after a few years out of the business and a bit rusty, I'm hoping someone can help me with the following question:

I have a textbook question that asks me to prove (A + !B).(!A + !B + C) = AC + !B using boolean algebra. I have verified this is correct quite easily using a truth table, but I'm struggling with the boolean algebra, and I can't seem to re-factor or minimise the L.H.S. satisfactorily, even after a couple of hours head scratching and doodling out smaller truth tables.

So far, using the distributive laws, I have come up with a broken down L.H.S.

A.!A + A.!B + AC + !A.!B + !B.!B + !B.C = AC + !B

I think A.!A resolves to 0, and !B.!B resolves to !B, so I think I am on the right track (as this gives me the AC + !B components of the R.H.S.), but I can't think where to go from here, particularly removing the !B.C component.

Alternatives I thought about were to convert the (!A + !B + C) to !(A .B) + C using De Morgans, or using inversion, but I think these would complicate the matter more than necessary. Am I using the distributive law correctly? Any hints on the right direction would be much appreciated.

Thanks and regards,

DMDog.