Given a logic module A that compares the magnitudes of two 3-bit numbers, X₃ = x₁x₂x₃ and Y₃ = y₁y₂y₃, where x₃ and y₃ are the least significant bits. Module A has two outputs G₃ and S₃, such that: G₃ = 1 iffX₃ > Y₃; S₃ = 1 iff X_{3 3}; and G₃ = S₃ = 0 iff X₃ = Y₃.

(a) Design a logic unit B such that together with module A it will serve as a comparator for two four-bit numbers, X₄ = x₁x₂x₃x₄ and Y₄ = y₁y₂y₃y₄, as shown in Fig. P5.9. Find expressions for G₄ and S₄ in terms of the inputs to unit B and show a realization of these expressions using only NAND gates.

(b) Show a realization of module A by means of only units of type B. Assume that the constants 0 and 1 are available.

Fig. P5.9