# Boolean algebra - dual and self dual functions

I'm having a problem understanding what influences the dual function has on the original function output.

For example, if I have a self dual function - for the input 11 the function out put will be 1, and the output of the dual function will be the same.

Now for dual function - I saw on the Internet that the dual function always give the opposite output compare to the original function. Im pretty confused because if we look on the function A*B Then the dual function is A+B. now for the input A=1,B=1 they giving the same output.

Some one can explain me what dual function is beside changing the operators?

Thanks!

Functions $f$ and $f_d$ are duals if

$$f(A,B,..) = \overline{f_d(\overline{A},\overline{B},..)}$$

or in other words if you invert all the inputs of a dual function (relative to the other dual), the output will also be inverted.

For example with the AND operation we have $f(A,B) = AB$ and so we have

$$\overline{f_d(\overline{A},\overline{B},..)} = AB$$ $$\implies f_d(\overline{A},\overline{B},..) = \overline{AB} = \overline{A}+\overline{B}$$ $$\implies f_d(A,B,..) = A + B$$

So the dual of the AND operation is the OR operation. Testing this out with $A=1$, $B=1$ we can see that $f(A,B) = AB = 1$ and $f_d(\overline{A},\overline{B}) = \overline{A}+\overline{B} = 0$ so the functions are infact duals.

It doesn't matter that the two functions will give the same output if the same input is given to both functions. The test you have to use to test if two functions are duals is to the invert the input into one dual function and check if the output is inverted relative to the other dual.

Below are images showing the implementation of the AND operation and it's dual (i.e the OR function), all thats been done to implement the dual is the inversion of inputs and the output.

The AND function: and it's dual (the OR function): 