I have a question regarding simplifying a circuit of a function below that has 5 logic gates in original.
$f = (A + B) \cdot (C + D) + (A + B) \cdot (C + D)' + C$
$= (A + B) \cdot ((C + D) + (C + D)') + C$
$= (A + B) \cdot 1 + \overline{C} \quad$ ($\overline{C}$ denotes the complement of $C$)
$= (A + B) + C$
Now, I have reduced to $2$ logic gates from $5$. But, here, am I allowed to change $(A + B) + C$ to $A + B + C$, so that I can reduce number of gates (i.e., $1$ logic gate) even more? If I am allowed, is there a name for this process (what kind of law is this)?
Thank you in advance.
Yes. You have correctly simplified the circuit by using in turn: (1) distribution, (2) disjunction of complements, (3) conjunction's identity.
Finally, you can say $\;(A+B)+C = A+B+C\;$ because of the associativity of disjunction.
Thus you have pared it down to just one logic gate, a triple-or gate (such as the one used in the original diagram),