Boolean Algebra · step 4 of 13

Boolean functions & truth tables

A Boolean function of n variables is a map {0,1}ⁿ → {0,1}, and it is completely determined by its 2ⁿ-row truth table. Since each row can independently output 0 or 1, there are 2^(2ⁿ) distinct functions of n variables — 16 for n = 2 and 256 for n = 3. Every one of them is expressible in sum-of-products form, which is why the table and the algebra carry the same information.

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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.

Enumerate the rows in binary orderFor n variables count from 0 to 2ⁿ − 1 in binary. This ordering makes K-map transfer and comparison straightforward.
Evaluate subexpressions in columnsOne column per operation, working outward from the innermost brackets.
Read an expression off the tableFor each row with output 1, write the product of literals describing that row; OR them together.
Count functions when asked2^(2ⁿ) total. Restricting to a property, such as monotone or self-dual, gives smaller counts.

Worked example

Build the truth table for f(x,y,z) = xy + z′ and write its sum-of-products form.

Eight rows. f = 1 when xy = 1, i.e. x = y = 1, or when z = 0.
Rows with z = 0: (0,0,0), (0,1,0), (1,0,0), (1,1,0) — all output 1.
Rows with z = 1: only (1,1,1) has xy = 1, so it outputs 1; the rest output 0.
Five rows output 1, giving five minterms.

Answer. f = x′y′z′ + x′yz′ + xy′z′ + xyz′ + xyz — which simplifies back to xy + z′.

Where marks get dropped

These are the specific errors that cost credit on boolean functions & truth tables questions — QED's rubric penalises each of them separately.

Miscounting rows. Three variables need eight rows; a seven-row table has an omission that invalidates everything after it.
Reading minterms with the wrong polarity. A row with x = 0 contributes x′, not x.
Confusing 2^(2ⁿ) with 2ⁿ. The number of functions is doubly exponential — 256 for three variables, not 8.

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Boolean functions & truth tables — frequently asked questions

How many Boolean functions of 2 variables are there?

2^(2²) = 2⁴ = 16, which is why the standard list of binary connectives — AND, OR, XOR, NAND, implication and so on — has exactly sixteen entries.

Is the sum-of-products form unique?

The canonical form built from minterms is unique. Simplified sum-of-products forms are not.

Can every Boolean function be written algebraically?

Yes — the minterm construction from the truth table always works, which proves {+, ·, ′} is functionally complete.

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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