How to Get a Simplified Sum of Products from a Truth Table

A truth table gives you the function completely. Turning it into a sum of products (SOP) expression takes two steps: write the canonical form, then simplify.

Step 1: the canonical SOP (sum of minterms)

For each row where the output $F$ is 1, write a minterm: the AND of every variable, where a variable appears plain if it is 1 in that row and complemented if it is 0. Then OR all the minterms.

Example truth table:

Row $m$	$A$	$B$	$C$	$F$
0	0	0	0	0
1	0	0	1	1
2	0	1	0	0
3	0	1	1	1
4	1	0	0	0
5	1	0	1	1
6	1	1	0	1
7	1	1	1	1

$F = 1$ in rows 1, 3, 5, 6 and 7, so $F = \sum m(1,3,5,6,7)$ and

$$F = A'B'C + A'BC + AB'C + ABC' + ABC.$$

This expression is correct but not minimal.

Step 2a: simplify with Boolean algebra

The key identity is adjacency: $XY + XY' = X(Y + Y') = X$. Two terms that differ only in one variable merge and lose that variable.

1. $A'B'C + A'BC = A'C$ (they differ only in $B$).
2. $AB'C + ABC = AC$.
3. $A'C + AC = C$.
4. $ABC' + ABC = AB$. Using $ABC$ a second time is allowed, because $X + X = X$.

So

$$F = C + AB.$$

Check: row 6 ($A=1, B=1, C=0$) gives $0 + 1 = 1$ ✓. Row 4 ($1,0,0$) gives $0 + 0 = 0$ ✓. Rows 0 and 2 give $0$ ✓.

Step 2b: simplify with a Karnaugh map

The map orders the columns in Gray code ($00, 01, 11, 10$), so neighbouring cells differ in one variable:

$A \backslash BC$	00	01	11	10
0	0	1	1	0
1	0	1	1	1

• The four 1s in columns $01$ and $11$ form a group where only $C = 1$ stays constant, giving $C$.
• The pair in row $A=1$, columns $11$ and $10$, gives $AB$.

That again yields $F = C + AB$. Rules: groups must be rectangles of size 1, 2, 4 or 8; make them as large as possible; overlapping is allowed; the map wraps around at the edges.

Don’t-care conditions

If some input combinations can never occur, mark them X. You may include them in a group if that makes the group larger, and otherwise ignore them. The calculator above accepts don’t-cares separately from the minterms.

For more than four variables

Karnaugh maps become unwieldy beyond four or five variables. The Quine–McCluskey method does the same merging systematically: list the minterms, repeatedly combine pairs that differ in one bit to find the prime implicants, then choose a minimal set that covers every minterm. This is what the calculator does.

Common mistakes

• Complementing the wrong way. A 0 in the row means the variable appears complemented in the minterm.
• Missing the wrap-around in a K-map: the first and last columns are adjacent.
• Stopping at non-maximal groups, which leaves extra literals.
• Confusing SOP with POS. For a product of sums, use the rows where $F = 0$, and write each maxterm as an OR with the opposite complement rule.