From truth tables to equations: minterms, maxterms, SOP and POS

Select exactly one truth-table row

Fix input order A, B, C with A most significant. Row 5 is 101, and $m_5=A\overline B C$ is 1 only on that row. A minterm contains every variable once; the name does not mean minimum gate count.

If F is 1 on rows 1, 3, 5, 6 and 7:

This canonical sum of products (SOP) ORs the one-row minterms. A general SOP need not include every variable in every term.

Select zero rows instead

The maxterm^{imaxterm An OR term containing every variable, false for exactly one input combination. AND together the maxterms of the false rows.} $M_2=A+\overline B+C$ is zero only on row 010. Pick each literal to be zero on that row. Since F is zero on rows 0, 2 and 4:

This is canonical product of sums (POS). Σm lists one rows; ΠM lists zero rows.

Canonical forms are a starting point

The same function simplifies to $F=C+AB$: C covers four rows, and AB covers the remaining 110. This is SOP but not canonical SOP. The next Karnaugh-map lesson makes such grouping visible. Always verify the simplified expression against all eight rows.

Lab · Connect three representations of one function

The A/B/C buttons change the input being observed. The cells change the function specification. The dashed outline marks the input row; blue borders mark the selected product term.

For the original function, set ABC=110. F remains 1 because AB is true even though C=0. Select the C and A·B groups and notice their shared m7 cell. Overlap does not change an OR result.

Click m6 twice: 1 → X → 0. The final function is just C. Row 6 leaves Σm and enters ΠM. The intermediate X means this row has no required output, not that a defined output may be ignored just to reduce cost.

Σm lists required ones, ΠM required zeros, and d unspecified rows. When d is nonempty, Σm or ΠM alone is not a complete truth-table specification. The lab minimizes product-term count first and literal count second, and shows one solution; this is not a claim about minimum physical gate area.