Question

How many Boolean functions on two independent Boolean variables a and b are dependent on either a or b or both?

Answer 1

Answer on Question #42787, Math, Abstract Algebra

Problem.

How many Boolean functions on two independent Boolean variables $a$ and $b$ are dependent on either $a$ or $b$ or both?

Solution.

There are $2^4 = 16$ different Boolean functions on two Boolean variables $a$ and $b$ (see table).

Two of these functions are independent of both $a$ and $b$ , as they are constants:

$f_{0}(a,b) = 0$

$f_{15}(a,b) = 1$

Four of these are functions of a single variable:

$f_{3}(a,b) = a$

$f_{5}(a,b) = b$

$f_{10}(a,b) = \overline{b};$

$f_{12}(a,b) = \overline{a}$

Ten other functions depend of both variables.

Answer: independent of both variables – 2 functions, dependent on either $a$ or $b - 4$ functions, dependent of both variables – 10 functions.

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