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S TLD K- MAP

Explore simplifying Boolean expressions using K-Maps and implementing them with NAND gates. Learn to reduce expressions, find prime implicants, and realize minimal expressions using basic gates.

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S TLD K- MAP

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  1. STLDK-MAP PREPAREDBY K.ASHOKKUMAR ASSISTANTPROFESSORDEPARTMENTOFECE

  2. M-Notations: Minterms and Maxterms

  3. Q. SimplifyF=Σm(0,1,4,5,8,9,10,11,14,15) Grouping: 1.Pair(2) 2.Quad(4) 3.Octet(8)

  4. Q. SimplifyF=Σm(0,1,4,5,8,9,10,11,14,15) CD AB 00 01 11 10 1 1 00 AICI 3 2 0 1 01 1 1 5 7 6 4 1 1 11 AC 12 13 15 14 1 1 1 1 10 8 9 11 10 ABI F=ABI+AC+AICI

  5. Q. F=AIBICIDI+AIBICDI+AIBCID+AIBCD+ABICIDI+ABICDI+ABCID+ABCD 00000010010101111000101011011111 m0m2m5m7m8m10m13m15 F=Σm(0,2,5,7,8,10,13,15) CD AB 00 01 11 10 1 1 00 3 2 0 1 01 1 1 5 7 6 4 BD 1 1 11 12 13 15 14 II BD 1 1 10 8 911 F=BD+BIDI 10

  6. SimplifythefollowingBoolean expressions usingK-Map andimplementthemusing NANDgates F(W,X,Y,Z)= XZ+W͛XY͛+WXY+W͛YZ+WY͛Z

  7. SimplifythefollowingBoolean expressions usingK-Map andimplementthemusing NANDgatesF(W,X,Y,Z)= XZ+W͛XY͛+WXY+W͛YZ+WY͛Z F(W,X,Y,Z)=XZ+W͛XY͛+WXY+W͛YZ+WY͛Z =XZ(W+WI)(Y+YI)+WIXYI(Z+ZI)+WXY(Z+ZI)+WIYZ(X+XI)+WYIZ(X+XI) =XZ(WY+WYI+WIY+WIYI)+WIXYIZ+WIXYIZI+WXYZ+WXYZI+WIXYZ+ WIXIYZ+WXYIZ+WXIYIZ =WXYZ+WXYIZ+WIXYZ+WIXYIZ+WIXYIZ+WIXYIZI+WXYZ+WXYZI+ WIXYZ+WIXIYZ+WXYIZ+WXIYIZ =WXYZ+WXYIZ+WIXYZ+WIXYIZ+WIXYIZI+WXYZI+WIXIYZ+WXIYIZ 1111 1101 0111 0101 0100 1110 0011 1001 m15m13 m7m5m4 m14 m3m9 F=Σŵ(3,4,5,7,9,13,14,15)

  8. F=Σŵ(3,4,5,7,9,13,14,15) EPI WXYZ 00 01 11 10 1 00 3 2 0 1 1 1 1 01 5 7 6 RPI 4 EPI 1 1 1 11 12 13 15 14 EPI 1 10 8 9 11 10 EPI 33

  9. F=Σŵ(3,4,5,7,9,13,14,15) YZ WIYZ WX 00 01 11 10 1 00 3 2 0 1 1 1 1 01 5 7 6 4 WIXYI 1 1 1 11 12 13 15 14 WXY 1 10 8 9 11 10 WYIZ F=WIXYI+WYIZ+WXY+WIYZ 34

  10. F=WIXYI+WYIZ+WXY+WIYZ 35

  11. ForthegivenfunctionT(w,x,y,z)= ∑(0,1,2,3,4,6,7,8,9,11,15) Showthemap Findallprimeimplicantsandindicatewhichareessential. FindaminimalexpressionforTandrealizeusingbasic gates.Is itunique? RPI yz wx 00 01 11 10 wIyI EPI 1 1 1 1 00 3 2 0 1 01 1 1 1 RPI 5 7 6 4 yzEPI 1 11 12 13 15 14 1 1 1 10 8 9 11 10 xIyI EPI RPI

  12. yz wx 00 01 11 10 wIz I EPI 1 1 1 1 00 3 2 0 1 01 1 1 1 5 7 6 4 yzEPI 1 11 12 13 15 14 1 1 1 10 8 9 11 10 xIyI EPI T(w,x,y,z)=xIyI+wIzI+yz

  13. Where Prime Implicants –PI Essential Prime Implicants –EPI Redundant Prime Implicants –RPI Grouping: 1.Pair(2) 2.Quad(4) 3.Octet(8)

  14. Prime Implicants –PI: Each square or Rectangle made up of the bunch of adjacent min terms is called a sub cube Each of these subs cubes is called a PI Essential Prime Implicants –EPI: The prime implicant which contains at least one ‘1’ which can not be covered by any other Prime implicant is called EPI Redundant Prime Implicants –RPI: The prime implicant whose ecah ‘1’ is covered at least by one EPI is called a RPI

  15. T(w,x,y,z)=xIyI+wIyI+yz KNIRANJANKUMARASST.PROFESSORPBRVITS

  16. Reduceusingmappingtheexpressionf=ΠM(2,8,9,10,11,12,14)andimplementusingReduceusingmappingtheexpressionf=ΠM(2,8,9,10,11,12,14)andimplementusing NORlogicgates

  17. Reduceusingmappingtheexpressionf=ΠM(2,8,9,10,11,12,14)andimplementusingReduceusingmappingtheexpressionf=ΠM(2,8,9,10,11,12,14)andimplementusing NORlogicgates CD AB 00 01 11 10 (B+CI+D) 0 00 3 2 0 1 01 (AI+D) 5 7 6 4 0 0 11 12 13 15 14 0 0 0 0 10 8 9 11 10 (AI+B) f=(AI+B)(AI+D)(B+CI+D)

  18. f=(AI+B)(AI+D)(B+CI+D)

  19. Q. Simplifythe givenexpression f(x,y,z)=∑(1,3,4,5) inPOSform f(x,y,z)=ΠM(0,2,6,7) yz x 01 11 00 10 (x+z) 00 0 0 1 3 2 0 0 1 4 5 7 6 (xI+yI) f(x,y,z)=(x+z)(xI+yI)

  20. Q.Simplifythegivenexpression f(x,y,z)=∑(0,2,6,7)using3variableK mapandimplement using(i)AND–NOR (ii)NAND-AND(iii)OR-NAND(iv)NOR-OR yz x 01 11 00 10 xIzI 01 1 0 1 3 2 1 1 1 4 5 7 6 xy f(x,y,z)=xy+xIzI

  21. f(x,y,z)=xy+xIzI (i)AND–NOR

  22. f(x,y,z)=xy+xIzI (ii)NAND-AND

  23. (iii)OR-NAND f(x,y,z)=xy+xIzI 45

  24. (iv)NOR-OR f(x,y,z)=xy+xIzI 46

  25. Minimise the following function and implement using AOI logic FunctionF(A,B,C,D)= Π(0, 2,4,8,9,12,14).

  26. Don’t care Combinations: Somelogiccircuitscanbedesignedsothattherearecertaininputconditionsforwhichtherearenospecifiedoutputlevels.Insuchcases,theoutputlevelsnotdefined,itcanbeEitherHIGHorLOW Don't cares in a Karnaugh map, or truth table, may be either 1s or 0s, as long as wedon't care what the output is for an input condition we never expect to see. ... When forming groups of cells, treat the don't care cell as either a 1 or a 0, or ignore the don't cares.

  27. F=Σm(0,2)+Σd(5,6,7) F=ΠM(1,3,4).Πd(5,6,7) F=Σm(0,2)+Σd(5,6,7)=ΠM(1,3,4).Πd(5,6,7)

  28. Q. Simplify F=Σm(0,1,4)+Σd(5,6,7) BC A 01 11 00 10 01 1 0 1 3 2 1 X X X 1 4 5 7 6 BI

  29. Q.SimplifyF=Σm(4,6,7,13,14)+Σd(5,10,12,15) CD AB 00 01 11 10 00 3 2 0 1 X 1 1 1 01 5 7 6 4 1 X X 1 11 12 13 15 14 B X 10 8 9 11 10 50

  30. Q. SimplifyF=Σm(0,2)+Σd(5,6,7) Q. SimplifyF(ABC)=Σm(0,1,3,7)+Σd(2,5) Q.Find the reduced SOP form of the following F(wxyz)=Σm(0,7,8,9,10,12)+Σd(2,5,13) Q. Reduce the following F(ABCD)=ΠM(0,3,4,7,8,10,12,14)+d(2,6)

  31. Q. SimplifyF=Σm(0,2,6,7) Q. Reduce the following F(xyz)=ΠM(0,2,5,7)

  32. 5VariableK-map CDE AB 000001011010110 111 101100 00 01 11 10 0 7 5 2 6 4 1 3 12 8 15 11 9 13 10 14 28 29 24 25 31 27 26 30 20 16 17 23 21 19 18 22 51

  33. A=0 A=1 01 DE DE 00 11 10 00 01 11 10 BC 00 BC 00 01 01 11 11 10 10 27/10/2016 52

  34. Q. Simplify F=Σm(0,2,3,4,6,7,9,11,16,18,19,20,22,23,25,27) BID II BE A=0 A=1 01 DE DE 00 11 10 00 01 11 10 BC 00 BC 00 1 1 1 1 1 1 17 19 18 16 1 3 2 0 1 1 1 1 1 1 01 01 20 23 22 21 4 7 6 5 11 11 28 29 31 30 12 13 15 14 1 1 25 1 1 10 10 24 27 26 8 9 11 10 BCIE F=BID+BIEI+BCIE

  35. Q. Reducethefollowingexpressiontothesimplest possiblePOSandSOPforms F=∑m(6,9,13,18,19,25,27,29,31)+∑d(2,3,11,15,17,24,28) BICID AIBIDEI SOPForm A=0 A=1 01 DE DE 00 11 10 00 01 11 10 BC 00 BC 00 X 1 1 X X 17 19 18 16 1 3 2 0 1 01 01 20 23 22 21 4 7 6 5 X 1 1 1 X 11 11 28 29 31 30 12 13 15 14 X 1 25 1 X 1 10 10 24 27 26 8 9 11 10 BE F=BE+BICID+AIBIDEI

  36. Q. Simplify F=ΠM(0,1,4,5,7,8,10,12,14,16,20,21,22,23,26,30).Π d(2,3,11,15,17,24,28) POSForm (B+D) (AI+B+CI) A=0 A=1 01 DE DE 00 11 10 00 01 11 10 BC 00 BC 000 X 0 X X 0 18 17 19 16 1 3 2 0 0 0 0 0 0 0 0 01 01 20 23 22 21 4 7 6 5 0 0 X X 0 11 11 28 29 31 30 12 13 15 14 0 26 X X 0 0 10 10 10 24 25 27 8 9 11 (BI+E) II (A+D+E) F27=/10(/2B01+6D)(BI+E)(A+D+E) (AI+B+CI) 56

  37. Ex-or Function:

  38. NAND Realisation

  39. NOR Realisation:

  40. Quine-McCluskey method (or) Tabular Minimization method (or) Prime implicants chart method

  41. Q.Simplify the following expression using Q-M Tabulation Method. • F(ABCD)= ∑m(0,2,3,6,7,8,10,12,13)

  42. Q.Simplify the following expression using Q-M Tabulation Method. F= ∑m(0,1,2,8,9,15,17,21,24,25,27,31)

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