Boolean Algebra · step 6 of 13

Sum-of-products & product-of-sums

The canonical sum-of-products (SOP) form ORs one minterm per row where the function is 1; the canonical product-of-sums (POS) form ANDs one maxterm per row where it is 0. Both are unique, and they are complementary — a function with k ones has k minterms and 2ⁿ − k maxterms. SOP maps directly to AND-OR circuits, POS to OR-AND circuits.

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Method: how to approach it

The order below is what examiners expect to see, and each step carries its own marks.

Build the truth table firstBoth forms are read directly off it, so an error there propagates everywhere.
Minterms from the 1-rowsFor each row output 1, write the product where a variable appears complemented iff it is 0 in that row. OR them.
Maxterms from the 0-rowsFor each row output 0, write the sum where a variable appears complemented iff it is 1 in that row — the opposite polarity to minterms. AND them.
Use Σ and Π notationf = Σm(1,3,5) lists the 1-rows by index; f = ΠM(0,2,4,6,7) lists the 0-rows. The index sets are complementary.

Worked example

For f(x,y) with f = 1 exactly on rows (0,1) and (1,0), give the canonical SOP and POS forms.

Row (0,1): x = 0, y = 1, so the minterm is x′y.
Row (1,0): the minterm is xy′. SOP = x′y + xy′.
0-rows are (0,0) and (1,1). For (0,0) the maxterm is (x + y).
For (1,1) the maxterm is (x′ + y′). POS = (x + y)(x′ + y′).

Answer. SOP: x′y + xy′. POS: (x + y)(x′ + y′). Both describe XOR.

Where marks get dropped

These are the specific errors that cost credit on sum-of-products & product-of-sums questions — QED's rubric penalises each of them separately.

Using the same polarity rule for maxterms as for minterms. Maxterms invert: a variable equal to 1 in the row appears complemented.
Mixing up which rows each form uses. SOP uses the 1-rows, POS uses the 0-rows.
Calling a simplified expression "canonical". Canonical forms include every variable in every term.

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Sum-of-products & product-of-sums — frequently asked questions

Why are canonical forms unique?

Because they are read directly off the truth table, and the table determines the function. Simplified forms lose that uniqueness.

How do the Σ and Π index sets relate?

They partition {0, …, 2ⁿ − 1}: every row is either a 1-row or a 0-row. So f = Σm(S) means f = ΠM(complement of S).

Which form gives a smaller circuit?

Whichever has fewer terms — if the function is 1 on only two rows, SOP is small; if it is 0 on only two, POS is. Comparing both is standard practice.

The rest of Boolean Algebra

Axioms, laws, simplification and Boolean functions. Each subtopic below has its own method, worked example and mark-losing traps.

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