it is perfectly valid to construct gates with more than two inputs so for example your and gate can have more than two inputs in this case you have three inputs can also have four 5 6 7 and so on uh you can also have your orgate with more than two inputs uh it's going to work in the similar it's going to work in the same way as your two input and gate and your two input orgate the principle Remains the Same so f is asserted only when a B and C are asserted that is for the and gate and for the orgate f is asserted if at least one of a b or c are asserted so let's draw the truth table uh let's call the output of the three input and gate as F and the output of the three input or gate as G and I'm going to show you how it looks like so a b c these are the three inputs have two outputs F which is a b c and G which is a or b or c so since you have three inputs you have eight combinations when all inputs are zeros okay now for the three input and gate the gate is asserted or the gate output is asserted only when all the three inputs are one and for the other cases it's going to be zero now for the three input R gate the gate is asserted if at least one of a b or c are asserted so except for this case the gate is asserted for the other combinations now let's take another example so we are again going to come up with the Boolean expression so here's the problem you have three inputs a b c all of them go into a three input and gate then you have a fourth input D it goes into a no gate uh for the no gate you cannot have more than two uh one input so that's a restriction on the no gate so the output of the and gate and the output of the not gate are fed into and orgate so what is the mathematical expression or the bullan expression for this one so it's going to be pretty much the same thing the same concept the output of this or the output over here is the output of the three input and gate which is A and B and C the output of the inverter is the inverse of D or the prime and this is your orgate so the output f equals A and B and C or D Prime