Sum-of-Products (SOP) Method

Digital Logic & Microprocessor — Digital Logic, NEC licence examination syllabus (Nepal Engineering Council).

Sum-of-Products (SOP) Method

Turning a table of what you want into a circuit that does it.

The problem this solves

You know what your circuit should do — you can write down, for every combination of inputs, what the output should be. That is a truth table, and it is easy to produce.

But you cannot build a truth table out of gates. You need an expression. SOP is the method that turns one into the other, and it is entirely mechanical.

The method, in one sentence

Write one term for every row that outputs 1, then OR all those terms together. That is the whole method — the rest is knowing how to write a single row as a term.

Writing one row as a term

For a row that outputs 1, ask: what would have to be true for the inputs to be exactly this?

✍️ Reading a row

Input is 1Write the letter as it is: A

Input is 0Write the letter with a bar: A̅ — meaning "A is NOT true"

Join them with ANDBecause ALL of them must hold at once for this row

So a row with A=0 and B=1 becomes A̅·B — "A is off and B is on". That single term is 1 for that row and 0 for every other, which is exactly what you want.

Worked example

Two rows give 1, so we write two terms and OR them:

Row A=0, B=1 → A̅·B Row A=1, B=0 → A·B̅ OUT = A̅·B + A·B̅

💡 That expression is XOR. Check it against the table if you like — it is 1 when the inputs differ and 0 when they match. This is a useful thing to notice, because recognising a familiar function can save you building it from scratch.

Why "sum of products"

The names are older than computers and come from ordinary algebra:

📖 The vocabulary

Productmeans AND — because AND behaves like multiplying: 1·1 = 1, and anything times 0 is 0.

Summeans OR — because OR behaves like adding: anything plus 1 gives 1.

So SOPis a sum (OR) of products (ANDs). The name is a description of the shape.

What it costs, and what comes next

SOP always works, and it is often wasteful. Ten rows outputting 1 gives ten AND terms and a very wide OR — a lot of silicon for something that may simplify enormously.

That is what Boolean algebra and Karnaugh maps are for. SOP gets you a circuit that is correct; those two get you one that is small. Always do them in that order.

Exam notes

📝 What gets asked

Table to expressionStraight application. Marks are for the correct bars — a 0 means barred, and getting that backwards loses the question.

Then simplifyUsually the second half of the same question, by K-map or Boolean laws.

Count your termsThe number of terms must equal the number of 1s in the output column. A quick check that catches dropped rows.

Syllabus points

Minterms; canonical & minimal SOP

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