Answer:
To prove that F = A’BC + A’BC’ + A’B’C + AB’C + ABC is equivalent to F = A'B + C, we can use proof by rewrite to simplify and transform one expression into the other.
Starting with F = A’BC + A’BC’ + A’B’C + AB’C + ABC, we can apply the distributive law to factor out A’ and get:
F = A’(BC + BC’ + B’C) + AB’C + ABC
Next, we can use the law of complementation to simplify A’(BC + BC’ + B’C) as A’B + A’C, and then use the distributive law again to factor out C:
F = A'B + A'C + AB'C + ABC
Finally, we can use the law of idempotence to simplify A'C + ABC as C + A'BC, and then substitute this into the above expression to get:
F = A'B + C
Therefore, we have shown that F = A’BC + A’BC’ + A’B’C + AB’C + ABC is equivalent to F = A'B + C using proof by rewrite. The key to the simplification was applying the distributive law, law of complementation, and law of idempotence, and using the fact that the uniting law allows us to combine terms that have a common variable.