--- title: "Minterms/maxterms; K-map Σm(0,1,2,5,8,14)+d, NOR" subject: "Digital Logic" chapter: 3 key: "semester_3/digital_logic:minterm-maxterm-kmap-nor-0125" source: https://www.bctengineer.com/q/semester_3/digital_logic/minterm-maxterm-kmap-nor-0125 site: BCT Engineering --- # Minterms/maxterms; K-map Σm(0,1,2,5,8,14)+d, NOR Digital Logic · Chapter 3: Combinational Logic Circuits - Asked in: 2076 Chaitra, 2+4+2 marks - Video: [Minimize the following expression using K map and realize using NOR gate](https://www.youtube.com/watch?v=2HpvaRRDPK8) — Maharshi Sanand Yadav T - 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-nor-0125 - Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3 ## Question Define minterms and maxterms. Simplify the following using k-map and implement the result using NOR gates only. F(A,B,C,D) = Σm(0,1,2,5,8,14) + d(4,10,13) ## Short answer Minterm: product of all variables, equals 1 for one combination. Maxterm: sum of all variables, equals 0 for one combination. Simplified F = A' C' + B' D' + A C D'. NOR-only: rewrite terms as (A+C)', (B+D)', (A'+C'+D)', feed to a NOR for F', then NOR inverter. Uses 7 NOR gates. ## 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 $F(A,B,C,D) = \Sigma m(0,1,2,5,8,14) + d(4,10,13)$ ```text CD AB 00 01 11 10 +---+---+---+---+ 00 | 1 | 1 | 0 | 1 | +---+---+---+---+ 01 | X | 1 | 0 | 0 | +---+---+---+---+ 11 | 0 | X | 0 | 1 | +---+---+---+---+ 10 | 1 | 0 | 0 | X | +---+---+---+---+ ``` - Quad m(0,1,4,5) → $\overline{A}\,\overline{C}$ - Quad m(0,2,8,10) → $\overline{B}\,\overline{D}$ - Pair m(10,14) → $AC\overline{D}$ $$ F = \overline{A}\,\overline{C} + \overline{B}\,\overline{D} + AC\overline{D} $$ Check: 0, 1, 5 by $\overline{A}\,\overline{C}$ (with 4); 2, 8 by $\overline{B}\,\overline{D}$ (with 10); 14 by $AC\overline{D}$ (with 10). Don't-care 13 is not used. (The minimal POS needs 4 terms, so the SOP is simpler.) #### NOR-only implementation For NOR-only realization of an SOP expression, write each product term as a NOR of complemented literals ($\overline{X}\,\overline{Y} = \overline{X+Y}$), feed these to a NOR gate (which gives $\overline{F}$), and invert with one more NOR whose inputs are tied. $$ \begin{aligned} \overline{A}\,\overline{C} &= \overline{A+C} \\ \overline{B}\,\overline{D} &= \overline{B+D} \\ AC\overline{D} &= \overline{\overline{A}+\overline{C}+D} \\ F &= \overline{\overline{P_1+P_2+P_3}} \end{aligned} $$ ```text A ──[NOR]── A' (inputs tied) C ──[NOR]── C' (inputs tied) A, C ──[NOR]── P1 ─┐ B, D ──[NOR]── P2 ─┤──[NOR]── F' A', C', D ──[NOR]── P3 ─┘ F' ──[NOR]── F (inputs tied) ``` Gates: 2 NOR inverters, two 2-input NORs, one 3-input NOR, a 3-input NOR giving $\overline{F}$ and a NOR inverter: 7 NOR gates. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. For ABC = 101, what is the maxterm M5? A. A B' C B. A' + B + C' C. A + B' + C D. A' B C' Answer: B. In a maxterm a variable is true if its value is 0 and complemented if 1, so 101 gives A' + B + C'. (Key point: Maxterm definition) 2. Which statement about minterms and maxterms is correct? A. A minterm is a sum term that is 0 for exactly one input combination B. A minterm is a product term equal to 1 for exactly one combination; a maxterm is a sum term equal to 0 for exactly one combination C. Both are 1 for exactly one combination D. A maxterm is a product term and its complement is a minterm of different variables Answer: B. Minterms are AND terms that are 1 for one combination; maxterms are OR terms that are 0 for one combination, with Mi = mi'. (Key point: Minterm vs maxterm) 3. What is the simplified SOP of Σm(0,1,2,5,8,14) + d(4,10,13)? A. A'C' + B'D' + ACD' B. A'C' + B'D' + BCD' C. A'B' + C'D' + ACD' D. A'C' + BD + ACD' Answer: A. Quads m(0,1,4,5) = A'C' and m(0,2,8,10) = B'D', plus pair m(10,14) = ACD'. (Key point: K-map simplified SOP) 4. How is a product term such as A'C' realised with NOR gates? A. As NOR(A', C') B. As NOR(A, C), since A'C' = (A + C)' C. As NAND(A, C) D. As NOR(A, C') Answer: B. X'Y' = (X + Y)', so a product of complemented literals is a NOR of the true literals; ACD' becomes NOR(A', C', D). 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