What Boolean function does the circuit below realize ?

Approach: use the 3-to-8 decoder outputs as minterms. The decoder inputs are (Z, Y, X) with outputs O0..O7 corresponding to minterms m0..m7.

Identify which decoder outputs are ORed (those make f = 1):

• When X = 1 and Z = 0 (Y can be 0 or 1): (Z,Y,X) = (0,0,1) → O1, (0,1,1) → O3

• When X = 0 and Z = 1 (Y can be 0 or 1): (Z,Y,X) = (1,0,0) → O4, (1,1,0) → O6

Therefore the ORed minterms are m1 + m3 + m4 + m6.

Express each pair and simplify:

• m1 + m3 = X Z' (Y' + Y) = X Z'

• m4 + m6 = X' Z (Y' + Y) = X' Z

Combine the terms to get f = xz' + x'z. This is the XOR of X and Z (true exactly when X ≠ Z), independent of Y.