A l- Mamoo n U n iv e rsit y C o l l ege Com p ute r Sc ie n ce Dep art m e n t Secon d Y ear

1. Al-MamoonUniversity College ComputerScienceDepartment SecondYear Architecture and LogicGates LogicGates By Dr.MazinAlzewary 2017-2018

2. BasicLogicGates • AND Gate • Symbol • Truth Table (T.T) thetabledescribe relation between inputs andoutputs (Gate function) • Math Expression Y=A.B • OR Gate • Symbol • Truth Table (T.T) • Math Expression Y=A+B • NOT Gate • Symbol • Truth Table (T.T) • Math Expression Y=A • Buffer Gate • Symbol • Truth Table (T.T)

3. Math expression Y=A • NANDGate • Symbol • Truth Table (T.T) • Math Expression Y=A.B • NOR Gate • Symbol • Truth Table (T.T) • Math Expression Y=A+B • XOR Gate • Symbol • Truth Table (T.T) • Math Expression Y=A+B • =A.B +A.B • XNOR Gate • Symbol

4. Truth Table (T.T) • Math expression Y=A+B • Universal NAND Universal NOR

5. BooleanExpression • Consist of logical (0 or1)variableA, B, C, etc and logical operations (throughgates) suchas * (AND), • + (OR), complement(NOT) • Minterms areAND terms with every variable present in eithertrueorcomplemented form such as A.B, B.C.D • Maxterms areOR terms with every variable intrueor complemented formsuch as (A+B), (B+C) • There aretwo typeof logical expression • Sum of Product (SOP) Y=A.B • Z=A.C +D X=ABC +E • Product of Sum (POS) M=(A+B)(B+C) • Q 1 Convert thelogical expression to logical gatesor to digital circuit • 1. Y=A.B+C 2. Y=A.B’ + B.C Q2)Convert thelogicalgates to Boolean expression 1. 2.

6. LogicalAlgebra(Rules) • Q1 Provethat A +AB =A • A+AB=A(1+B) • =A.1 =A • Q2 Provethat A +AB=A +B • =(A+AB)+AB • =(AA+AB)+AB • =AA+AB+AA+AB • =(A+A)(A+B) • =A+B • Q3 Provethat (A + B) (A +C)=A+AB • =AA+AC+AB+BC • =A+AC+AB+BC • =A(1+C)+AB+BC • =A*1+AB+BC • =A(1+B)+BC • =A*1+BC • =A+BC • BooleanExpression Simplification • SimplificationUsingBoolean Algebra (Rules) • Order of execution is • Parentheses • NOT

7. AND • OR • Q1) Simplifythe expression usinglogical rules Y=ABC • Q2)Simplifythe expression usinglogical rules Y=AB(C+D) Q3)Simplifythe expression usinglogical rules Y=(A+B+C)D Q4)Simplifythe expression usinglogical rules Y=[AB(C+BD)+AB]C=(ABC+ABBD+AB)C =(ABC+0+AB)C =(ABC+AB)C =ABCC+ABC =ABC+ABC =BC(A+A) =BC.1 =BC Q5)Simplifythe expression usinglogical rules and draw circuit AB+AC+ABC

8. 2. SimplificationUsing Karnaugh Map (K-MAP) Karnaugh Map • Takeeach mintermfromexpression and find thesuitable cell in theMAP (AB, AB, etcfor two variable ) and write1 forthat location • Write 0to the rest location • Build block forthe onessuchthat • Contiguouscells • Theblock is 2,4,8,etc location • Choosebigger block first • Thecell (1)could be contain in morethan oneblock • Build Boolean term for each block • Thesimplified expression is the sum of all sub expression • Q1)Simplifyusingkarnaugh map • 1. Y= 2. Y=

9. Q2) Reducethe expressionusingK-MAP X=ABC+ABC+ABC+ABC+ABC Using SOP POS a. b. Q3) Reducethe expressionusingK-MAP Y=ABCD+ABCD+ ABCD+ABCD+ ABCD+ABCD+ABCD using1. SOP 2 POS 1.

10. 2. LogicFunction Representation 1. LogicFunction as Boolean Expression Form Y=AB+AB+AB Y=ABC +ABC +ABC +ABC +ABC And these could be solved usingK-Map But if wehaveY=AB+ABC+ABC Theterm ABnot includeall threevariable and how could weinsert it in the map IfwemultiplytheABby(C+C) to get AB=AB*1=AB*(C+C)=ABC +ABC Then the newexpression could be Y=ABC +ABC +ABC +ABD And could be solved usingK-Map 9

11. 2. Logic Function asTruth TableForm X(A,B,C)= (1,4,5,6,7) Y=ABC +ABC +ABC +ABC +ABC Thestandard mapforanyfunction could be mansion bellow

12. Q)Simplifythe followingusingK-Map X(A,B,V,D)= (1,4,8,9,11,12,13,15) Don’tCare Thereis situation that the value ofcell not affect the expression( thesameif it is 0 or 1)this situation is called don’t care(X) andcould be used as 0 or 1 in buildingtheblock (thefunction do not care foroutput 0 or1) Q)Simplifythe Booleanfunction given in the truth table

17. Questions Q1)Simplifythe followingBooleanexpression usingKarnaugh map Y=AB+AB+AB Y=ABC +ABC +ABC +ABC +ABC +ABC Y=ABC +ABC +ABC +ABC +ABC +ABC Y=ABCD +ABCD +ABCD +ABCD + ABCD+ABCD Y=AB+ABC +ABC +ABC +ABC +ABC Y =A+ABC + ABC +ABC +ABC +ABC +ABC Y=ABC +BC +C+A+ABC +ABC Q2)Convert the followingSOP expression to an equivalent POS expression. Q3)Determinethe values of A, B, C, and Dthat makethe sum term equal to zero. Q4) Whichof thefollowingexpressionsisinthesum-of-products (SOP) form? (A+B)(C +D) (A)B(CD) AB(CD) AB+CD Q5)DerivetheBoolean expressionfor thelogiccircuitshownbelow:

18. Q6)From thetruthtablebelow,determinethestandardSOPexpression. Q7) OneofDe Morgan'stheoremsstatesthat differencebetween: .Simplystated, thismeansthatlogicallythereisno aNOR and anAND gatewithinvertedinputs aNAND andanOR gatewithinvertedinputs anAND and aNOR gatewithinvertedinputs aNOR and aNAND gatewithinvertedinputs Q8) ApplyingDeMorgan'stheorem totheexpression A. , weget_. B. C. D. Q9) Whichoutputexpression might indicateaproduct-of-sumscircuitconstruction? A. B. C. D. Q10) For theSOPexpression A. 1 , howmany1sareinthetruthtable'soutputcolumn?

19. Q11) Howmanygateswouldbe requiredtoimplement thefollowingBooleanexpressionbeforesimplification?XY + X(X+Z)+ Y(X+Z) Q12) Determinethevaluesof A,B,C,and D thatmaketheproduct term A. A=0,B=1,C =0,D =1 B. A=0,B=0,C =0,D =1 C. A=1,B=1,C =1,D =1 D. A=0,B=0,C =1,D =0 Q13) AC +ABC =AC A. True equal to1. B. False Q14)Applyingthedistributivelaw totheexpression A. , weget. B. C. D.