Write out the truth table for the 4-line-to-2-line encoder that takes a four-line decimal signal and converts it to binary code. Design, and draw, the circuit to implement this encoder.

Question

Asked on June 28, 2026Computer Science

This computer science problem involves algorithmic thinking and programming concepts. The solution below explains the approach, logic, and implementation step by step.

Accepted Answer · 2026-06-28T10:00:55.714Z

Here's how to solve question 4b: Step 1: Identify the original AND-OR-INVERT function and apply De Morgan's theorem. The standard AND-OR-INVERT (AOI) function with four inputs A, B, C, D is given by: Y = (A · B) + (C · D) Applying De Morgan's theorem, X+Z = X · Z: Y = (A · B) · (C · D) Applying De Morgan's theorem again to each term, X · Z = X + Z: Y = (A + B) · (C + D) This is the alternative form using the four input variables in their complemented form. The alternative form is: Y = (A + B) · (C + D) Step 2: Draw the logic circuit for the original AND-OR-INVERT function. The circuit for Y = (A · B) + (C · D) consists of two AND gates, one OR gate, and one NOT gate (represented by the bubble on the OR gate output). ` A ---| | | AND |--- B ---| | | | OR |--- O--- Y C ---| | | | | AND |--- D ---| | ` Step 3: Draw the logic circuit for the alternative form. The circuit for Y = (A + B) · (C + D) consists of four NOT gates, two OR gates, and one AND gate. ` A ---O---| | B ---O---| OR |--- | AND |--- Y C ---O---| | | D ---O---| OR |--- ` Step 4: Construct truth tables for both functions to prove their equivalence. The truth table for Y_1 = (A · B) + (C · D) and Y_2 = (A + B) · (C + D) is shown below. The columns for Y_1 and Y_2 are identical, proving their equivalence. |c|c|c|c|c|c|c|c|c|c|c|c|c|c|c| A & B & C & D & A · B & C · D & (A · B)+(C · D) & Y_1 & A & B & C & D & A+B & C+D & Y_2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ That's 2 down. 3 left today — send the next one.

Accepted Answer · 2026-06-28T10:00:55.714Z

Here's how to solve question 4b: Step 1: Identify the original AND-OR-INVERT function and apply De Morgan's theorem. The standard AND-OR-INVERT (AOI) function with four inputs A, B, C, D is given by: Y = (A · B) + (C · D) Applying De Morgan's theorem, X+Z = X · Z: Y = (A · B) · (C · D) Applying De Morgan's theorem again to each term, X · Z = X + Z: Y = (A + B) · (C + D) This is the alternative form using the four input variables in their complemented form. The alternative form is: Y = (A + B) · (C + D) Step 2: Draw the logic circuit for the original AND-OR-INVERT function. The circuit for Y = (A · B) + (C · D) consists of two AND gates, one OR gate, and one NOT gate (represented by the bubble on the OR gate output). ` A ---| | | AND |--- B ---| | | | OR |--- O--- Y C ---| | | | | AND |--- D ---| | ` Step 3: Draw the logic circuit for the alternative form. The circuit for Y = (A + B) · (C + D) consists of four NOT gates, two OR gates, and one AND gate. ` A ---O---| | B ---O---| OR |--- | AND |--- Y C ---O---| | | D ---O---| OR |--- ` Step 4: Construct truth tables for both functions to prove their equivalence. The truth table for Y_1 = (A · B) + (C · D) and Y_2 = (A + B) · (C + D) is shown below. The columns for Y_1 and Y_2 are identical, proving their equivalence. |c|c|c|c|c|c|c|c|c|c|c|c|c|c|c| A & B & C & D & A · B & C · D & (A · B)+(C · D) & Y_1 & A & B & C & D & A+B & C+D & Y_2 \\ 0 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 1 & 1 & 1 & 1 \\ 0 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 1 & 1 & 0 & 1 & 1 & 1 \\ 0 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 1 & 0 & 1 & 1 & 1 & 1 \\ 0 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 \\ 0 & 1 & 0 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 1 & 1 & 1 & 1 \\ 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 & 1 & 1 & 1 \\ 0 & 1 & 1 & 0 & 0 & 0 & 0 & 1 & 1 & 0 & 0 & 1 & 1 & 1 & 1 \\ 0 & 1 & 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 \\ 1 & 0 & 0 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 1 & 1 & 1 & 1 \\ 1 & 0 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 1 & 1 & 0 & 1 & 1 & 1 \\ 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 & 1 & 1 & 1 & 1 \\ 1 & 0 & 1 & 1 & 0 & 1 & 1 & 0 & 0 & 1 & 0 & 0 & 1 & 0 & 0 \\ 1 & 1 & 0 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 1 & 0 & 1 & 0 \\ 1 & 1 & 0 & 1 & 1 & 0 & 1 & 0 & 0 & 0 & 1 & 0 & 0 & 1 & 0 \\ 1 & 1 & 1 & 0 & 1 & 0 & 1 & 0 & 0 & 0 & 0 & 1 & 0 & 1 & 0 \\ 1 & 1 & 1 & 1 & 1 & 1 & 1 & 0 & 0 & 0 & 0 & 0 & 0 & 0 & 0 \\ That's 2 down. 3 left today — send the next one.

Write out the truth table for the 4-line-to-2-line encoder that takes a four-line decimal signal and converts it to binary code. Design, and draw, the circuit to implement this encoder.

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Write out the truth table for the 4-line-to-2-line encoder that takes a four-line decimal signal and converts it to binary code. Design, and draw, the circuit to implement this encoder.

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