--- title: "Minterms/maxterms; K-map Σm(0,1,2,4,6,8,9,12,13,14)" subject: "Digital Logic" chapter: 3 key: "semester_3/digital_logic:minterm-maxterm-kmap-01246" source: https://www.bctengineer.com/q/semester_3/digital_logic/minterm-maxterm-kmap-01246 site: BCT Engineering --- # Minterms/maxterms; K-map Σm(0,1,2,4,6,8,9,12,13,14) Digital Logic · Chapter 3: Combinational Logic Circuits - Asked in: 2081 Bhadra, 2+4 marks - Video: [Simplification of Boolean expressions using Kmap.](https://www.youtube.com/watch?v=-GFnvD-RogA) — Engg-Course-Made-Easy - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/digital_logic/minterm-maxterm-kmap-01246 - Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3 ## Question Define minterms and maxterms. Simplify the following using k-map. F(A,B,C,D) = Σm(0,1,2,4,6,8,9,12,13,14) ## Short answer A minterm is an AND term with all variables, true for one input; a maxterm is an OR term with all variables, false for one input. K-map simplification yields F = A'D' + BD' + AC' + B'C'. ## Answer #### Minterms and maxterms A **minterm** is a product (AND) term that contains every variable of the function exactly once, in true or complemented form. It equals 1 for exactly one input combination. A variable is written true if its value is 1 and complemented if it is 0; e.g. for ABC = 101 the minterm is $m_5 = A\overline{B}C$. A function in **canonical SOP** is a sum of minterms: $F = \Sigma m(\dots)$. A **maxterm** is a sum (OR) term that contains every variable exactly once, in true or complemented form. It equals 0 for exactly one input combination. A variable is written true if its value is 0 and complemented if it is 1; e.g. for ABC = 101 the maxterm is $M_5 = \overline{A}+B+\overline{C}$. A function in **canonical POS** is a product of maxterms: $F = \Pi M(\dots)$. $M_i = \overline{m_i}$, and a function's minterm list and maxterm list are complementary (together they hold all $2^n$ numbers). #### K-map simplification $F(A,B,C,D) = \Sigma m(0,1,2,4,6,8,9,12,13,14)$ ```text CD AB 00 01 11 10 +---+---+---+---+ 00 | 1 | 1 | 0 | 1 | +---+---+---+---+ 01 | 1 | 0 | 0 | 1 | +---+---+---+---+ 11 | 1 | 1 | 0 | 1 | +---+---+---+---+ 10 | 1 | 1 | 0 | 0 | +---+---+---+---+ ``` **Grouping the 1s (SOP):** - Quad m(0,2,4,6) → $\overline{A}\,\overline{D}$ - Quad m(4,6,12,14) → $B\overline{D}$ - Quad m(8,9,12,13) → $A\overline{C}$ - Quad m(0,1,8,9) → $\overline{B}\,\overline{C}$ $$ F = \overline{A}\,\overline{D} + B\overline{D} + A\overline{C} + \overline{B}\,\overline{C} $$ Check: 0, 2, 4, 6 by $\overline{A}\,\overline{D}$; 12, 14 by $B\overline{D}$; 8, 9, 12, 13 by $A\overline{C}$; 0, 1, 8, 9 by $\overline{B}\,\overline{C}$. All 10 minterms are covered by four quads. **Grouping the 0s (POS):** the 0s are at 3, 5, 7, 10, 11, 15. - Pair of 0s M(5,7) → $(A+\overline{B}+\overline{D})$ - Pair of 0s M(10,11) → $(\overline{A}+B+\overline{C})$ - Quad of 0s M(3,7,11,15) → $(\overline{C}+\overline{D})$ $$ F = (\overline{C}+\overline{D})(A+\overline{B}+\overline{D})(\overline{A}+B+\overline{C}) $$ Both forms use 8 literals; either is an acceptable simplified answer. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In a maxterm, how is a variable written? A. True if its value is 1 B. True if its value is 0 and complemented if 1 C. Always complemented D. Always true Answer: B. The maxterm is 0 at its own combination, so a 0 variable appears true and a 1 variable complemented. (Key point: Maxterm writing rule) 2. Which term does the quad m(4,6,12,14) give? A. BD' B. AD' C. A'D' D. B'D' Answer: A. Cells 4,6,12,14 have B=1 and D=0, so BD'. (Key point: Quad for minterms 4,6,12,14) 3. What is the minimal SOP of Σm(0,1,2,4,6,8,9,12,13,14)? A. A'D' + BD' + AC' + B'C' B. A'D' + BD' + AB C. C' + D' D. A'D' + B'C' Answer: A. Four quads: A'D', BD', AC' and B'C'. (Key point: SOP result) 4. Which cells are grouped to obtain the POS form? A. The 1s B. The 0s: 3,5,7,10,11,15 C. The don't-cares D. Cells 0 and 15 Answer: B. POS groups the 0s; here quad M(3,7,11,15) = C' + D' etc. (Key point: POS grouping) ## Related in Digital Logic - [Minterms/maxterms; K-map Σm(0,1,2,5,8,14)+d, NOR](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-nor-0125) — Digital Logic, chapter 3 - [Minterms/maxterms; K-map Σm(2,4,5,13,14)+d, universal gates](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-universal) — Digital Logic, chapter 3 - [Minterms/maxterms; K-map Σm(0,1,2,4,5,9,12,15)+d, NAND](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-nand-0124) — Digital Logic, chapter 3 - [Minterms/maxterms; K-map Σ(1,3,5,8,10,11,12)+d](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Aminterm-maxterm-kmap-1358) — Digital Logic, chapter 3, asked 2× - [K-map Σm(7,9,12,13,14,15)+d, NOR-only](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-nor-sum-7-9-12) — Digital Logic, chapter 3 - [K-map Σ(1,2,3,8,9,10,11,13,14)+d, POS](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-pos-form-12389) — Digital Logic, chapter 3