Each of the following actually represents a set of four assignments corresponding to the possible assignments to the input variables: $$f_1(w,x,y,z) = (1,3,5,7,9,11,12,14)$$ $$f_2(w,x,y,z) = (0,2,4,5,8,9)$$ Find/Find out: a. How many functions does $$f_1$$ represent? b. How many functions does $$f_2$$ represent?

Answer:

a. How many functions does $$f_1$$ represent?

The function $$f_1(w,x,y,z)$$ is a function of four variables: w, x, y, and z. Each of these variables can take on two values (0 or 1), so there are $$2^4 = 16$$ possible combinations of inputs.

The function $$f_1$$ is defined for 8 of these combinations, so it represents 8 different functions.

To see this, consider the following table:

w	x	y	z	f_1(w,x,y,z)
0	0	0	0	1
0	0	0	1	3
0	0	1	0	5
0	0	1	1	7
0	1	0	0	9
0	1	0	1	11
0	1	1	0	12
0	1	1	1	14

Each row in this table represents a different function, so there are 8 functions in total.

b. How many functions does $$f_2$$ represent?

The function $$f_2(w,x,y,z)$$ is also a function of four variables: w, x, y, and z. Each of these variables can take on two values (0 or 1), so there are $$2^4 = 16$$ possible combinations of inputs.

The function $$f_2$$ is defined for 6 of these combinations, so it represents 6 different functions.

To see this, consider the following table:

w	x	y	z	f_2(w,x,y,z)
0	0	0	0	0
0	0	0	1	2
0	0	1	0	4
0	0	1	1	5
0	1	0	0	8
0	1	0	1	9

Each row in this table represents a different function, so there are 6 functions in total.