# How can you obtain SOP and POS (Boolean Logic) given the truth table?

Recently, it appears to me that few people on here are having issues determining SOP (Sum-of-Products) and POS (Product-of-Sums). So lets go through an example to help those people solve their future homework questions instead of giving them the answer. The initial question concerning SOP and POS is how you can obtain it from a simple K-map or truth table of few inputs Boolean logic?

What are the steps to take to solve for SOP? Or represent the truth table as SOP?

What are the steps to take to solve for POS? Or represent the truth table as POS?

Given the following truth table:

First of all, these are the steps you should follow in order to solve for SOP:

1. Write AND terms for each input combination which produce HIGH output.
2. Write the input variable if it is 1, and write the complement if the variable value is 0.
3. OR the AND terms to obtain the output function. In other words, add the AND terms together to produce your wanted output function.
4. SOP will have this form from the truth table given in the question: $$F = \overline{A}BC + A\overline{B}C + AB\overline{C} + ABC$$

The first term: $$\overline{A}BC$$ A is equal to 0 in the truth table when output F is equal to 1. The second term: $$A\overline{B}C$$ B is equal to 0 in the truth table when output F is equal to 1. The third term: $$AB\overline{C}$$ C is equal to 0 in the truth table when output F is equal to 1. The fourth term: $$ABC$$ A, B, C are all equal to 1 in the truth table when output F is equal to 1.

Secondly, these are the steps you should follow in order to solve for POS:

1. Write OR terms when the output F is equal to 0.
2. Write the input variable (A, B, C) if the value is zero, and write the complement if the input is 1.
3. AND the OR terms to obtain the output function. In other words, multiple the OR terms together to get the final output logic equation.
4. POS will have this form from the truth table given in the question: $$F=(A+B+C)(A+B+\overline{C})(A+\overline{B}+C)(\overline{A} + B + C)$$

The first term: $$(A+B+C)$$ A, B, and C are equal to zero and the output F is equal to zero as well. The second term: $$(A+B+\overline{C})$$ Output F is equal to zero but C is equal to 1. Hence why we take complement. The third term: $$(A+\overline{B}+C)$$ Output F is equal to zero but B is equal to 1. Hence why we take complement. The fourth term: $$(\overline{A}+B+C)$$ Output F is equal to zero but A is equal to 1. Hence why we take complement.

Essentially, you have to follow the three first steps in order to successfully for SOP and POS. You could have a lot more terms depending on your output results or the number of inputs you have.

Note that the results shown above are not in the reduced format. You could potentially reduce those Boolean-logic equations much more.