![Page 1: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/1.jpg)
Simplification of Expression
By: Ali Mustafa
![Page 2: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/2.jpg)
How to simplify the Boolean Function
• Algebric method
• K-Map method (Karnaugh)
• Quine – Mc Cluskey method
• VEM (Variable entered mapping)
![Page 3: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/3.jpg)
Algebraic Forms( A way to represent any function)
Sum of Products (SOP)
Product of Sums (POS)
![Page 4: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/4.jpg)
Sum of Products (SOP)
• Switching functions formed by SUMMING (ORing) PRODUCT (ANDed) terms.
![Page 5: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/5.jpg)
Sum-of-products(SOP)
• Product terms are known as minterm
• Output = 1 , for SOP
![Page 6: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/6.jpg)
Sum-of-products (cont.)• ANDed product–input combination for which output
is true
• Each variable appears exactly once, in true or inverted form (but not both)
![Page 7: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/7.jpg)
Sum-of-products
Example : Simplify the following three variable boolean expression Y = ∑ m (2,4,6)
Solution: y = m2 +m4 + m6
=A’BC’+AB’C’+ABC’=A’BC’+AC’(B’+B)
=A’BC’+AC’= C’(A’B + A)
Hint =(A’+ A = 1) = C’(A’+ A) (B + A)
= C’(B + A)
![Page 8: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/8.jpg)
How to build a circuit from the (SOP)function?
F = AB'C + A'B’C + AB'C + ABC' + ABC
Answer: SOP AND/OR Two-level Implementation
![Page 9: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/9.jpg)
Product of Sums (POS)
• Switching functions formed by taking the PRODUCT (ANDing) of SUM (ORed) terms.
![Page 10: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/10.jpg)
Product-of-sums
• Sum terms are known as Maxterm
• Output = 0 , POS
![Page 11: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/11.jpg)
Product-of-sums (cont’d)• Sum term (or maxterm)
• ORed sum of literals –input combination for which output is false
• Each variable appears exactly once, in true or inverted form (but not both)
![Page 12: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/12.jpg)
Product-of-sums (Self Task)
• Example : Simplify the following three variable boolean expression Y = ∏ M (1,3,5)
?
![Page 13: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/13.jpg)
How to build a circuit from (POS) function?
F(A, B, C) = (A + B + C) (A + B' + C) (A + B’ + C’)
Answer
POS: OR/AND Two-level Implementation
![Page 14: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/14.jpg)
SOP and POS Represent the same function
![Page 15: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/15.jpg)
POS vs. SOP
• Any expression can be written either way
• Can convert from one to another using theorems
• Sometimes SOP looks simpler
AB + CD = ( A + C )( B + C )( A + D )( B + D )
• Sometimes POS looks simpler
(A + B)(C + D) = BD + AD + BC + AC
• SOP will be most commonly used
![Page 16: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/16.jpg)
Standard or canonical SOP & POS forms
Sr # Logical Expression Type of Expression
1 Y = AB + ABC’ + A’BC NON STANDARD
2 Y = AB + A’B + A’B’ STANDARD
3 Y = (A’ + B)(A + B)(A + B’) STANDARD
4 Y = (A’ + B)(A + B +C) NON STANDARD
![Page 17: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/17.jpg)
Conversion Procedure for Standard SOP
1. For Each term ,we find the missing literal
2. Then, we AND term with the term formed by ORing the missing literal and its compliment
Y = AB + AC’ + BC
![Page 18: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/18.jpg)
Conversion Procedure for Standard SOPExample: Convert the expression into the standard SOP form
Y = AB + AC’ + BC
Solution: Given expression Y = AB + AC’ + BC
Step 1: Find the missing literals
Y = AB + AC’ + BC
Step 2: AND each term with (Missing literal + its compliment)
Y = AB(C + C’) + AC’(B + B’) + BC(A +A’)
Simplify the expression to get the standard SOP form
Y = ABC + ABC’ + ABC’+ AB’C’ + ABC + A’BC
Y = ABC + ABC’+ AB’C’ + A’BC(Each term consists of all literal)
C B A
A + A = A
![Page 19: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/19.jpg)
Conversion Procedure for Standard POS• Example: Convert the expression into the standard SOP form
Y = ( A+B )( A+C )( B +C’)
Step 1: Find the missing literals
Step 2: OR each term with (Missing literal + its compliment)
Y = ( A + B + CC’ )( A+ BB’ +C )( AA’ + B +C’ )
Y =( A + B + C )( A + B + C’ ) ( A+ B +C )( A+ B’ +C ) ( A + B +C’ )( A’ + B +C’ )
Y =( A + B + C )( A + B + C’ )( A+ B’ +C )( A’ + B + C’ )
![Page 20: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/20.jpg)
SELF TASKS
• Convert the following expressions into standard SOP & POS forms
1. Y = AB + AC + BC
2. Y = (A + B) (B’ + C )
3. Y = A + BC + ABC
![Page 21: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/21.jpg)
Boolean Expression from a Logic Circuit
• To derive the Boolean expression for a given logic circuit, begin at the left-most inputs and work toward the final output, writing the expression for each gate.
![Page 22: Simplification of Expression - Digital Logic Design (EEE 241)digitallogicdesign.weebly.com/uploads/1/3/5/4/13541180/... · 2018. 10. 4. · Simplification of Expression By: Ali Mustafa](https://reader034.vdocuments.us/reader034/viewer/2022051918/600a820ddc2757271414d663/html5/thumbnails/22.jpg)
Example