--- title: "K-map and universal gate: xy + x'z + yz" subject: "Basic Electronics Engineering" chapter: 6 key: "semester_2/basic_electronics_engineering:kmap-universal-gate-xy" source: https://www.bctengineer.com/q/semester_2/basic_electronics_engineering/kmap-universal-gate-xy site: BCT Engineering --- # K-map and universal gate: xy + x'z + yz Basic Electronics Engineering · Chapter 6: Digital Electronics - Asked in: 2071 Magh, 3+2 marks - Video: [Boolean Expression Solving using K-Map](https://www.youtube.com/watch?v=ewkhVRIzsN0) — TutorialsPoint - 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-universal-gate-xy - Whole chapter: https://www.bctengineer.com/md/semester_2/basic_electronics_engineering/chapter-6 ## Question Simplify the following Boolean expression using K-map and realize it by using universal gate of your interest. F(x, y, z) = xy + x'z + yz ## Short answer F = xy + x'z (yz is redundant). Realized with NAND gates: double inversion gives F = ((xy)'.(x'z)')'. Circuit uses four NANDs: one for xy, one inverter (NAND) for x', one for x'z, and one to combine. ## Answer F(x,y,z) = xy + x'z + yz. Minterms: xy = m6 + m7, x'z = m1 + m3, yz = m3 + m7. So F = Σm(1, 3, 6, 7). #### K-map ```text yz 00 01 11 10 x = 0 | 0 | 1 | 1 | 0 | x = 1 | 0 | 0 | 1 | 1 | ``` Groups: (m6, m7) = xy, and (m1, m3) = x'z. The group (m3, m7) = yz is redundant (it is the consensus term). **Simplified: F = xy + x'z** #### Realization using NAND gates Using double inversion: F = ((xy + x'z)')' = ((xy)' · (x'z)')' ```text x ------------+ [NAND 1]--- (xy)' ------+ y ------------+ | [NAND 4]--- F = xy + x'z x ---[NAND 2]--- x' --+ | (inputs tied) [NAND 3]--- (x'z)' z --------------------+ ``` - NAND 1: NAND(x, y) = (xy)' - NAND 2 (used as NOT, inputs joined): gives x' - NAND 3: NAND(x', z) = (x'z)' - NAND 4: NAND of the NAND 1 and NAND 3 outputs gives F = xy + x'z Total: 4 two-input NAND gates (the universal gate used). ## Check yourself The answer's key points as multiple-choice questions, with the correct option and why. 1. xy + x'z + yz equals: A. Sigma m(1, 3, 6, 7) B. Sigma m(1, 3, 5, 7) C. Sigma m(2, 3, 6, 7) D. Sigma m(1, 2, 6, 7) Answer: A. The answer states this: Sigma m(1, 3, 6, 7). (Key point: Minterms) 2. Why is yz (m3, m7) redundant? A. It covers a 0 B. Its minterms are already covered by xy and x'z (consensus term) C. It is not a valid group D. It is the same as xy Answer: B. The answer states this: Its minterms are already covered by xy and x'z (consensus term). (Key point: Redundant term) 3. The simplified function is: A. F = xy + yz B. F = x'z + yz C. F = xy + x'z D. F = xy + x'z + yz Answer: C. The answer states this: F = xy + x'z. (Key point: Simplified function) 4. The NAND realization of xy + x'z uses: A. 3 two-input NAND gates B. 5 two-input NAND gates C. 2 two-input NAND gates D. 4 two-input NAND gates Answer: D. The answer states this: 4 two-input NAND gates. (Key point: NAND gate count) ## 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 - [Verify Boolean identities](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Averify-boolean-identities) — Basic Electronics Engineering, chapter 6 - [K-map: two 3-variable functions](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-two-expressions) — Basic Electronics Engineering, chapter 6 - [K-map: A'B + BC' + AC'](https://www.bctengineer.com/question.md?key=semester_2%2Fbasic_electronics_engineering%3Akmap-abc-simple) — Basic Electronics Engineering, chapter 6, asked 2× - [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 ## Related in other subjects - [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 - [K-map Σ(0,1,4,8,10,11,12)+d, realize circuit](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Akmap-realize-circuit-01248) — Digital Logic, chapter 3, asked 2× - [Directional derivative of xy²+yz³](https://www.bctengineer.com/question.md?key=semester_2%2Fmathematics%3Adirectional-derivative-xy2-yz3) — Mathematics, chapter 5, asked 4× - [De Morgan; X-NOR using a universal gate](https://www.bctengineer.com/question.md?key=semester_3%2Fdigital_logic%3Ademorgan-xnor-universal-gate) — Digital Logic, chapter 2