Due Wednesday 28 Sept. 2005 in class at beginning of class in drop box.
In the following table ! preceeding an input specifies NOT
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Laws Identity Null Itempotent Inverse Commutative Associative Distributive Absorption De Morgan's Cancelation |
AND 1*A=A 0*A=0 A*A=A A*!A=0 A*B=B*A (A*B)*C=A*(B*C) A+B*C=(A+B)*(A+C) A*(A+B)=A !(A*B)=!A+!B !!A=A |
OR 0+A=A 1+A=1 A+A=A A+!A=1 A+B=B+A (A+B)+C=A+(B+C) A*(B+C)=A*B+A*C A+A*B=A !A*!B=!(A+B) |
General rule: When given a combination of ANDs, ORs and NOTs, NOTS are applied to following input first, next statements in parens are resolved, next NOTs applied to parens, then ANDs, and finally ORs
Part 1 Absorption 5 points
Prove the AND form of absorption using the other rules. Do NOT use the OR form of absorption as a shortcut.
Part 2 Simplification 5 points each
For each of the following problems, simplify the statements using boolean algebra.
Example:
Problem: F=S*T+V*W+R*S*T
| F=S*T+R*S*T+V*W | - Commutative law |
| F=1*S*T+R*S*T+V*W | - Identity law |
| F=(1+R)*S*T+V*W | - Distributave(or) |
| F=1*S*T+V*W | - Null |
| F=S*T+V*W | - Identity law |
Simplify the equation. Show your work and name the rules as they are applied. You may use any rule except absorption and you don't have to state the commutative and associative rules when applied.
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A. F=A*B*C+D*E+E B. F=A*(B+A+C) C. F=(A*B+C+D*E)*E*D D. F=!(!D*!D*E) |
E. F=C*(A+B+!(!C+!D))*D F. F=(A*C+B+D)*B G. F=A*C+C(!A+A*B) H. F=(A+B*C*D+E)*E*D |
I. F=(A+B)(A+C)(!A+C) J. C=!(A*B)*(A+B) K. D=(A*B)+C*(!(A*B)*(A+B)) L. D=!(!(A*B)*(A+B)*C)*(!(A*B)*(A+B)+C) |
Simplify the following statements. Show your work and name the rules as they are applied. DeMorgan and cancellation may be useful. Draw the truth table for each of these. Make sure both original and simplified versions give same results. The link below will give you a prepared set of truth tables for the full expressions. Add your simplified tables to the bottom of the page.
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A. A+!A+B B. A*(!A+B) C. !A+A*B |
D. !A+A*!B E. A*(!A*B) F. !A*(A+!B) |
A. Draw a 2 unit adder using full adder modules (Show the gates). This adder will be able to add 2 2-bit numbers. It should also be able to take a carry from a less significant bit pair adder module and pass the carry on the next higher signifcant adder unit.
B. Draw a 6 selection decoder.
C. Draw the gates for the Part 2. E. problem. Use original (starting) statement.
D. Draw the XOR as a combination of ANDs, NANDs, ORs, and NORs.