--- title: "K-map: two 3-variable functions" subject: "Basic Electronics Engineering" chapter: 6 key: "semester_2/basic_electronics_engineering:kmap-two-expressions" source: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/kmap-two-expressions site: BCT Engineering --- # K-map: two 3-variable functions Basic Electronics Engineering · Chapter 6: Digital Electronics - Asked in: 2072 Magh, 3+3 marks - Video: [3 Variable Karnaugh Map Tutorial](https://www.youtube.com/watch?v=A2Kypcw7EMk) — Learn Electronics Online - Answer checked by Claude Opus 5.5 (high effort) on 2026-10-14 - Page: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/kmap-two-expressions - Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-6 ## Question Simplify the following expression: i) F(x,y,z) = xyz + x'y'z + xy'z' + x'y'z' + x'yz ii) F(x,y,z) = Σ(0,2,5,6) ## Short answer i) F = Σ(0,1,3,4,7) simplifies to x'y' + y'z' + yz. ii) F = Σ(0,2,5,6) simplifies to x'z' + yz' + xy'z. ## Answer #### i) F = xyz + x'y'z + xy'z' + x'y'z' + x'yz Minterms: xyz = 7, x'y'z = 1, xy'z' = 4, x'y'z' = 0, x'yz = 3. So F = Σm(0, 1, 3, 4, 7). ```text yz 00 01 11 10 x = 0 | 1 | 1 | 1 | 0 | x = 1 | 1 | 0 | 1 | 0 | ``` Groups: - (m0, m1): x'y' - (m0, m4): y'z' (same column, x varies) - (m3, m7): yz **F = x'y' + y'z' + yz** (an equally simple form is x'z + y'z' + yz; m4 can only be covered by y'z'). #### ii) F = Σm(0, 2, 5, 6) ```text yz 00 01 11 10 x = 0 | 1 | 0 | 0 | 1 | x = 1 | 0 | 1 | 0 | 1 | ``` Groups: - (m0, m2): x = 0, z = 0 gives x'z' - (m2, m6): y = 1, z = 0 gives yz' - m5 alone: xy'z **F = x'z' + yz' + xy'z** ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. In part i, the function is: A. Sigma m(0, 1, 3, 4, 7) B. Sigma m(0, 1, 3, 5, 7) C. Sigma m(1, 3, 4, 6, 7) D. Sigma m(0, 2, 3, 4, 7) Answer: A. The answer states this: Sigma m(0, 1, 3, 4, 7). (Key point: Minterms of part i) 2. The pair (m2, m6) gives: A. x'z' B. xz' C. yz' D. xy Answer: C. The answer states this: yz'. (Key point: Part ii group) 3. For Sigma m(0, 2, 5, 6), the result is: A. F = x'z' + yz' B. F = x'z' + xy'z C. F = x'y' + yz + xz D. F = x'z' + yz' + xy'z Answer: D. The answer states this: F = x'z' + yz' + xy'z. (Key point: Part ii answer) ## Related in Basic Electronics Engineering - [K-map: x'yz + x'y'z + xyz with gates](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-xyz-realize-gates) — Basic Electronics Engineering, chapter 6, asked 2× - [K-map: 3-variable five-term expression](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-xyz-five-term) — Basic Electronics Engineering, chapter 6 - [K-map and universal gate: xy + x'z + yz](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-universal-gate-xy) — Basic Electronics Engineering, chapter 6 - [K-map: 4-variable minterm list](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-4var-sigma) — Basic Electronics Engineering, chapter 6 - [Simplify two expressions and draw circuits](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Asimplify-and-draw-circuits) — Basic Electronics Engineering, chapter 6 ## Related in other subjects - [CNF: S→XYZ|XY|aZ](https://www.bctengineer.com/question.md?key=semester_3%2Ftheory_of_computation%3Acnf-conversion-xyz) — Theory of Computation, chapter 3 - [Directional derivative of xy²+yz³](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adirectional-derivative-xy2-yz3) — Mathematics, chapter 5, asked 4× - [Testing lossless functions; Foster I and Cauer II](https://www.bctengineer.com/question.md?key=semester_5%2Ffilter_design%3Alossless-test-foster-i-cauer-ii) — Filter Design, chapter 4