logic diagrams!

[t.man]

how can one go about in representing the following Boolean function using NAND gates with a minimum number of gates?

z = A' BC + AB'C + ABC' + ABC

 

[MrAl]

Hi,

One way is to reduce the equation using Boolean Algebra.

Another way is to generate a truth table and look at the
pattern of 1's and 0's to look for a simpler logic representation.

Example using Boolean Algebra:
z = ABC' + ABC
reduces to:
z = AB + AB
because it doesnt matter whether C is high or low, so it gets eliminated,
and of course that reduces to:
z = AB
so
z = ABC' + ABC = AB
and that would be implemented using one two-input 'AND' gate.

 

[moghli_amateur]

originally posted byt.man
how can one go about in representing the following Boolean function using NAND gates with a minimum number of gates?

z = A' BC + AB'C + ABC' + ABC

Typically logic designers use K-map but in this particular case the boolean laws can be easily used.
z= A' BC + AB'C + AB
z= A' BC +A(B'C+B)
z=A'BC+A(B+C)
z=A'BC+AB+AC
z=B(A'C+A)+AC
z=AB+BC+AC
As you can see this is just a minimal expression in Sum of Products form.
Since "AND-OR" combination is equivalent to "NAND-NAND" combination, you can relaise the above expression by three 2 input nanad gate and one three input nand gate.

Pessisim never won a war

 

[Chaerl]

With this boolean equation (which is correctly derived), we can do 6 NAND gates to get the right circuit. I hope I'm right.

 

[Krumlink]

AND each logic input and OR each group together.
z = A' BC + AB'C + ABC' + ABC

 

[blueroomelectronics]

Whoa long time no see Krumlink

 

[Chaerl]

AND each logic input and OR each group together.
z = A' BC + AB'C + ABC' + ABC

In this case, you got it wrong. It should be the minimum number of NAND. In this case, we must simplify first the equation with the result of z=AB+BC+AC.
Afterwards, we can represent each the AND and OR with its equivalent NAND circuit. To further lessen the circuit, we must simplify more the equation by grouping. Thus, z=AB+(BC+AC). You can choose which term you want to group. I'll try to attach the circuit next time.

 

[Hero999]

AND each logic input and OR each group together.
z = A' BC + AB'C + ABC' + ABC

Welcome back.