Question

We briefly discussed Shannon's expansion theorem in class which states that a general Boolean function F(a₁,a₂,...aₙ) can be formulated by subfunctions. For example, F can be expanded in terms of two variables (i,j) as follows: F(a₁,a₂,...aₙ) = F(a₁,a₂,...aᵢ = 0, aⱼ = 0,...,aN) a'ᵢa'ⱼ + F(a₁,a₂,...aᵢ = 0, aⱼ = 1,...,aN) a'ᵢa'ⱼ + F(a₁,a₂,...aᵢ = 1, aⱼ = 0,...,aN) a'ᵢa'ⱼ + F(a₁,a₂,...aᵢ = 1,aⱼ = 1,...,aN) a'ᵢa'ⱼ

​Using this concept, implement the function F=A′C′D+BCD′+ABD′ using a single 4-to-1 MUX (whose selectors will be S₁ = C,S₀ = D), a single 2 input AND gate and as many inverters as needed.

Answer 1

Using selectors C and D, a single 4-to-1 multiplexer can implement the Boolean function F=A'C'D+BCD'+ABD' by mapping the terms of the function to the multiplexer inputs, requiring only one inverter and not using the 2-input AND gate.

Step-by-step explanation:

We will implement the given Boolean function F=A'C'D+BCD'+ABD' using a single 4-to-1 multiplexer, a single 2-input AND gate, and as many inverters as needed. The selectors will be S1 = C and S0 = D.
First, we must express the function in terms of the selectors C and D to align with the inputs of the multiplexer. The multiplexer inputs (I0, I1, I2, I3) will be related to the combinations of C and D as follows: C' D' (I0), C' D (I1), C D' (I2), C D (I3). Now, map the terms of the function to these inputs.

• I0: Only the term A'C'D applies. As C and D are both 0, we get simply A' for this input.
• I1: No terms apply when C is 0 and D is 1, so this input is 0.
• I2: The term BCD' applies here. With C as 1 and D as 0, we use B for this input.
• I3: The term ABD' applies here. Since D is 1, we ignore this term, and we are left with no terms for C and D both being 1, so this input is 0.

An inverter is needed on the A input for I0. No additional inverters are needed for B. The AND gate is not used because the final MUX implementation only requires the inputs I0 and I2 and no combined terms are necessary.