What are the canonical forms of Boolean expressions?

Concept

A canonical form is a single, unique, fully-expanded expression for a Boolean function in which every term lists all the input variables. Because the function's truth table is fixed, each function has exactly one such expression in each canonical form, making the representation standard and unambiguous.

The two canonical forms

The naming of each form follows directly from how its terms are built, not from an arbitrary label. A minterm is an AND term (a product term) that evaluates to 1 for exactly one row of the truth table; a maxterm is an OR term (a sum term) that evaluates to 0 for exactly one row. ORing together — i.e. summing — the minterms of the rows where the function is 1 is therefore called the Sum of Minterms; ANDing together — i.e. multiplying — the maxterms of the rows where the function is 0 is therefore called the Product of Maxterms. 'Sum' pairs with minterms because minterms are exactly the AND-terms defined for the 1-rows, and 'product' pairs with maxterms because maxterms are exactly the OR-terms defined for the 0-rows.

• Sum of Minterms (SOM): OR together one minterm for every truth-table row where the function equals 1; each minterm is an AND of all variables (in true or complemented form).

• Product of Maxterms (POM): AND together one maxterm for every row where the function equals 0; each maxterm is an OR of all variables.

Applying it to the choices

The question asks for the names of these two forms. Contrasting each offered pair by what it actually denotes:

• OR and XOR: single-gate operations, each evaluated on one pair of signals (e.g. 1 OR 0 = 1; 1 XOR 1 = 0).

• NOR and XNOR: also single-gate operations, each evaluated on one pair of signals (e.g. 0 NOR 0 = 1; 1 XNOR 0 = 0).

• Sum of Minterms and Product of Maxterms: whole-table expressions, each defined over every row of the truth table.

• "More than one of the above": would require at least two of the other option values to each independently satisfy the question — combining multiple values together.

• "None of the above": would require that not one of the other option values satisfies the question — rejecting the entire set at once.

Cross-check

SOM corresponds to the canonical Sum-of-Products (Σ of minterms) and POM to the canonical Product-of-Sums (Π of maxterms) — the two standard normal forms in Boolean algebra. So the pair Sum of Minterms and Product of Maxterms is the correct answer; the 'none of the above' and 'more than one' choices are ruled out because that pair is present and valid.