Boolean Algebra · step 7 of 13

Karnaugh maps & minimal expressions

A Karnaugh map rearranges a truth table into a grid whose rows and columns are labelled in Gray code, so physically adjacent cells differ in exactly one variable. Grouping adjacent 1s in blocks of size 1, 2, 4, 8 eliminates the variables that change within the block, and the largest legal groups give the minimal sum-of-products form.

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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.

Label in Gray codeColumns run 00, 01, 11, 10 — not binary order. This is what makes neighbouring cells differ in one bit.
Group in powers of twoBlocks must be rectangular with side lengths that are powers of two. Groups may overlap, and the map wraps around all edges.
Make groups as large as possibleA group of 2ᵏ cells eliminates k variables. Bigger groups always give shorter terms.
Cover every 1 at least oncePrefer essential prime implicants — 1s coverable by only one maximal group — then cover the remainder with the fewest additional groups.

Worked example

Minimise f(x,y,z) = Σm(0,1,2,3,5) with a K-map.

Minterms 0–3 are exactly the cells with x = 0, forming a group of four.
That group eliminates y and z, leaving the term x′.
Minterm 5 is x=1, y=0, z=1. It pairs with minterm 1 (x=0,y=0,z=1), a group of two.
That pair eliminates x, leaving y′z.

Answer. f = x′ + y′z, using one group of four and one of two.

Where marks get dropped

These are the specific errors that cost credit on karnaugh maps & minimal expressions questions — QED's rubric penalises each of them separately.

Labelling the map in binary order 00, 01, 10, 11. Adjacent cells then differ in two bits and every grouping is invalid.
Forgetting the wrap-around. The leftmost and rightmost columns are adjacent, and so are the top and bottom rows — corner cells can form a group of four.
Using groups of size 3 or 6. Only powers of two are legal.

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Karnaugh maps & minimal expressions — frequently asked questions

How large can K-maps get?

Four variables is comfortable, five and six are possible with layered maps, and beyond that Quine–McCluskey or software is used. The visual adjacency stops helping.

What is a prime implicant?

A group that cannot be enlarged. An essential prime implicant covers at least one 1 that no other prime implicant covers, so it must appear in every minimal form.

Can I use K-maps for POS?

Yes — group the 0s instead and read the result as a product of sums, complementing the variable polarities. That is the dual procedure.

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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