--- title: "K-map Σ(1,2,3,8,9,10,11,13,14)+d, POS" subject: "Digital Logic" chapter: 3 key: "semester_3/digital_logic:kmap-pos-form-12389" source: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-pos-form-12389 site: BCT Engineering --- # K-map Σ(1,2,3,8,9,10,11,13,14)+d, POS Digital Logic · Chapter 3: Combinational Logic Circuits - Asked in: 2072 Chaitra, 4+3 marks - Video: [SOP AND POS WITH K-MAP - Minimize SOP and POS with K-map solved examples - Hindi](https://www.youtube.com/watch?v=i5QjUVgY-XU) — Electronics Subjectified - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_3/digital_logic/kmap-pos-form-12389 - Whole chapter: https://www.bctengineer.com/md/semester_3/digital_logic/chapter-3 ## Question Simplify Σ(1,2,3,8,9,10,11,13,14) + d(0,4,7,12) by using K-Map and write its standard product of sum (POS) expression. ## Short answer Grouping 1s yields F = B' + A C' + A D'. Grouping 0s using don’t-cares 4 and 7 as 0 gives F = (A + B')(B' + C' + D'). The standard POS is the product of maxterms where F=0: Π M(4,5,6,7,15). ## Answer Given $F(A,B,C,D) = \Sigma m(1,2,3,8,9,10,11,13,14) + d(0,4,7,12)$. The 0-cells are 5, 6, 15. #### K-map ```text CD AB 00 01 11 10 +---+---+---+---+ 00 | X | 1 | 1 | 1 | +---+---+---+---+ 01 | X | 0 | X | 0 | +---+---+---+---+ 11 | X | 1 | 0 | 1 | +---+---+---+---+ 10 | 1 | 1 | 1 | 1 | +---+---+---+---+ ``` #### Simplified SOP (grouping 1s) - Octet m(0,1,2,3,8,9,10,11) → $\overline{B}$ - Quad m(8,9,12,13) → $A\overline{C}$ - Quad m(8,10,12,14) → $A\overline{D}$ $$ F = \overline{B} + A\overline{C} + A\overline{D} $$ #### Simplified POS (grouping 0s) Treat don't-cares 4 and 7 as 0 so the 0s form large groups: - Quad of 0s M(4,5,6,7) → $(A+\overline{B})$ - Pair of 0s M(7,15) → $(\overline{B}+\overline{C}+\overline{D})$ $$ F = (A+\overline{B})(\overline{B}+\overline{C}+\overline{D}) $$ Check: cell 5 (A=0, B=1) makes $A+\overline{B} = 0$; cell 15 (B=C=D=1) makes the second factor 0. All 1-cells give both factors = 1. #### Standard POS expression The standard (canonical) POS is the product of the maxterms where F = 0. For the simplified POS, the 0s are cells 4, 5, 6, 7 (from $A+\overline{B}$) and 7, 15 (from $\overline{B}+\overline{C}+\overline{D}$): $$ F = \Pi M(4,5,6,7,15) $$ $$ \begin{aligned} F = {} & (A+\overline{B}+C+D)(A+\overline{B}+C+\overline{D})(A+\overline{B}+\overline{C}+D) \\ & (A+\overline{B}+\overline{C}+\overline{D})(\overline{A}+\overline{B}+\overline{C}+\overline{D}) \end{aligned} $$ In terms of the given data only, $F = \Pi M(5,6,15)\cdot d(0,4,7,12)$; the simplification chose don't-cares 4 and 7 as 0. ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. To get the simplified POS, which cells are grouped? A. The 1-cells B. The 0-cells (and useful don't-cares) C. Only don't-cares D. Only cells 5 and 15 Answer: B. POS is obtained by grouping 0s; each group gives a sum term with complemented sense. (Key point: Which cells to group for POS) 2. What is the simplified POS of this function? A. (A + B')(B' + C' + D') B. (A' + B)(B + C + D) C. B' + AC' + AD' D. (A + B)(C + D) Answer: A. Quad M(4,5,6,7) = A + B' and pair M(7,15) = B' + C' + D'. (Key point: Simplified POS) 3. Which don't-cares are taken as 0 in the POS simplification? A. 0 and 12 B. 4 and 7 C. 7 and 12 D. 0 and 4 Answer: B. Setting d4 and d7 to 0 makes the quad of 0s M(4,5,6,7) and the pair M(7,15). (Key point: Don't-care use) 4. What is the standard POS of the simplified function? A. ΠM(5,6,15) B. ΠM(4,5,6,7,15) C. ΠM(0,4,7,12) D. ΠM(1,2,3) Answer: B. The simplified POS has 0s at 4,5,6,7 and 15, so F = ΠM(4,5,6,7,15). 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