Design the circuit with a PLA having three inputs and two outputs.i) F1 (A, B, C) = ∑m (1, 2, 6, 7) ii) F2 (A, B, C) = ∑(1, 3, 5, 7)

1. Break Down the Problem

We need to design a Programmable Logic Array (PLA) with the following specifications:

• Three inputs: A, B, C
• Two outputs: F1 and F2
• The function expressions for F1 and F2 based on the given minterms.

2. Relevant Concepts

A PLA consists of:

• An AND plane: to generate product terms.
• An OR plane: to combine product terms into outputs.

We will represent the functions using minterms:

• $F_1(A, B, C) = \Sigma m(1, 2, 6, 7)$
• $F_2(A, B, C) = \Sigma m(1, 3, 5, 7)$

3. Analysis and Detail

Step 1: List the Minterms

For 3 inputs (A, B, C), the minterms are:

• Minterm 0: ABC = 000
• Minterm 1: ABC = 001
• Minterm 2: ABC = 010
• Minterm 3: ABC = 011
• Minterm 4: ABC = 100
• Minterm 5: ABC = 101
• Minterm 6: ABC = 110
• Minterm 7: ABC = 111

Step 2: Generate the Product Terms

Using the minterms for $F_1$ and $F_2$:

• For F1:

  • $m(1) = A'B'C$
  • $m(2) = A'BC'$
  • $m(6) = AB'C$
  • $m(7) = ABC$

  Thus, $F_1 = A'B'C + A'BC' + AB'C + ABC$

• For F2:

  • $m(1) = A'B'C$
  • $m(3) = A'BC$
  • $m(5) = AB'C$
  • $m(7) = ABC$

  Thus, $F_2 = A'B'C + A'BC + AB'C + ABC$

Step 3: Simplify Functions

• Function F1: $F_1 = A'B'C + A'BC' + AB'C + ABC$ cannot be simplified further without additional tools.

• Function F2: $F_2 = A'B'C + A'BC + AB'C + ABC$

4. Verify and Summarize

Both functions $F_1$ and $F_2$ can be implemented in the PLA. We can draw the corresponding AND and OR planes.

Final Answer

The PLA has the following connections:

1. AND Plane Outputs:

   • Product terms for F1: $P_1 = A'B'C, P_2 = A'BC', P_3 = AB'C, P_4 = ABC$
   • Product terms for F2: $P_1 = A'B'C, P_2 = A'BC, P_3 = AB'C, P_4 = ABC$

2. OR Plane Connections:

   • $F_1 = P_1 + P_2 + P_3 + P_4$
   • $F_2 = P_1 + P_2 + P_3 + P_4$

This configuration successfully implements the given functions on a PLA with 3 inputs and 2 outputs.