Cpr E 281 Digital Logic Instructor Alexander Stoytchev
- Slides: 88
Cpr. E 281: Digital Logic Instructor: Alexander Stoytchev http: //www. ece. iastate. edu/~alexs/classes/
Boolean Algebra Cpr. E 281: Digital Logic Iowa State University, Ames, IA Copyright © Alexander Stoytchev
Administrative Stuff • HW 1 is due today
Administrative Stuff • HW 2 is out • It is due on Wednesday Sep 6 @ 4 pm. • Submit it on paper before the start of the lecture
Boolean Algebra • An algebraic structure consists of § a set of elements {0, 1} § binary operators {+, } § and a unary operator { ’ } or { } • Introduced by George Boole in 1854 George Boole 1815 -1864 • An effective means of describing circuits built with switches • A powerful tool that can be used for designing and analyzing logic circuits
Axioms of Boolean Algebra 1 a. 0 0 = 0 1 b. 1 + 1 = 1 2 a. 1 1 = 1 2 b. 0 + 0 = 0 3 a. 0 1 = 1 0 = 0 3 b. 1 + 0 = 0 + 1 = 1 4 a. If x=0, then 4 b. If x=1, then x=1 x=0
Single-Variable Theorems 5 a. x 0 = 0 5 b. x + 1 = 1 6 a. x 1 = x 6 b. x + 0 = x 7 a. x x = x 7 b. x + x = x 8 a. x 8 b. x + 9. x=x x=0 x=1
Two- and Three-Variable Properties 10 a. x y = y x 10 b. x + y = y + x Commutative 11 a. x (y z) = (x y) z 11 b. x + (y + z) = (x + y) + z Associative 12 a. x (y + z) = x y + x z 12 b. x + y z = (x + y) (x + z) Distributive 13 a. x + x y = x 13 b. x (x + y) = x Absorption
Two- and Three-Variable Properties 14 a. x y + x 14 b. (x + y) (x + y=x y) = x 15 a. x y = x + y 15 b. x + y = x y Combining De. Morgan’s theorem 16 a. x + x y = x + y 16 b. x (x + y) = x y 17 a. x y + y z + x z = x y + x z 17 b. (x+y) (y+z) (x+z)=(x+y) (x+z) Consensus
Now, let’s prove all of these
The First Four are Axioms (i. e. , they don’t require a proof) 1 a. 0 0 = 0 1 b. 1 + 1 = 1 2 a. 1 1 = 1 2 b. 0 + 0 = 0 3 a. 0 1 = 1 0 = 0 3 b. 1 + 0 = 0 + 1 = 1 4 a. If x=0, then 4 b. If x=1, then x=1 x=0
But here are some other ways to think about them
0 0 AND gate 0 1 1 1 OR gate
0 0 0 AND gate x 1 0 0 1 1 x 2 0 1 f 0 0 0 1 1 OR gate x 1 0 0 1 1 x 2 0 1 f 0 1 1 1
0 0 0 AND gate x 1 0 0 1 1 x 2 0 1 f 0 0 0 1 1 OR gate x 1 0 0 1 1 x 2 0 1 f 0 1 1 1
1 1 AND gate 1 0 0 0 OR gate
1 1 1 AND gate x 1 0 0 1 1 x 2 0 1 f 0 0 0 1 0 0 0 OR gate x 1 0 0 1 1 x 2 0 1 f 0 1 1 1
0 1 1 0 0 1 1
0 1 1 0 AND gate x 1 0 0 1 1 x 2 0 1 f 0 0 0 1 0 1 0 0 1 1 OR gate x 1 0 0 1 1 x 2 0 1 f 0 1 1 1
0 1 1 0 AND gate x 1 0 0 1 1 x 2 0 1 f 0 0 0 1 0 1 0 0 1 1 OR gate x 1 0 0 1 1 x 2 0 1 f 0 1 1 1
0 1 1 0 AND gate x 1 0 0 1 1 x 2 0 1 f 0 0 0 1 0 1 0 0 1 1 OR gate x 1 0 0 1 1 x 2 0 1 f 0 1 1 1
0 1 1 0
0 1 NOT gate 1 0 NOT gate x x 0 1 1 0
Single-Variable Theorems 5 a. x 0 = 0 5 b. x + 1 = 1 6 a. x 1 = x 6 b. x + 0 = x 7 a. x x = x 7 b. x + x = x 8 a. x 8 b. x + 9. x=x x=0 x=1
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately.
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0 0 = 0 // axiom 1 a
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0 0 = 0 // axiom 1 a ii) If x = 1, then we have 1 0 = 0 // axiom 3 a
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0+1 = 1 // axiom 3 b
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0+1 = 1 // axiom 3 b ii) If x = 1, then we have 1+1 = 1 // axiom 1 b
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0 1 = 0 // axiom 3 a ii) If x = 1, then we have 1 1 = 1 // axiom 2 a
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0 1 = 0 // axiom 3 a ii) If x = 1, then we have 1 1 = 1 // axiom 2 a
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0+0 = 0 // axiom 2 b ii) If x = 1, then we have 1+1 = 1 // axiom 1 b
The Boolean variable x can have only two possible values: 0 or 1. Let’s look at each case separately. i) If x = 0, then we have 0+0 = 0 // axiom 2 b ii) If x = 1, then we have 1+1 = 1 // axiom 1 b
i) If x = 0, then we have 0 0 = 0 ii) // axiom 1 a If x = 1, then we have 1 1 = 1 // axiom 2 a
i) If x = 0, then we have 0 0 = 0 ii) // axiom 1 a If x = 1, then we have 1 1 = 1 // axiom 2 a
i) If x = 0, then we have 0+0 = 0 ii) // axiom 2 b If x = 1, then we have 1+1 = 1 // axiom 1 b
i) If x = 0, then we have 0+0 = 0 ii) // axiom 2 b If x = 1, then we have 1+1 = 1 // axiom 1 b
i) If x = 0, then we have 0 1 = 0 ii) // axiom 3 a If x = 1, then we have 1 0 = 0 // axiom 3 a
i) If x = 0, then we have 0 1 = 0 ii) // axiom 3 a If x = 1, then we have 1 0 = 0 // axiom 3 a
i) If x = 0, then we have 0+1 = 1 ii) // axiom 3 b If x = 1, then we have 1+0 = 1 // axiom 3 b
i) If x = 0, then we have 0+1 = 1 ii) // axiom 3 b If x = 1, then we have 1+0 = 1 // axiom 3 b
i) If x = 0, then we have x = 1 // axiom 4 a let y = x = 1, then we have y=0 // axiom 4 b Therefore, x = x (when x =0)
ii) If x = 1, then we have x = 0 // axiom 4 b let y = x = 0, then we have y=1 // axiom 4 a Therefore, x = x (when x =1)
x y y x
x y y x AND gate x 0 0 1 1 y 0 1 f 0 0 0 1 OR gate x 0 0 1 1 y 0 1 The order of the inputs does not matter. f 0 1 1 1
x y z 0 0 0 1 1 1 0 0 1 1 1 x y z Truth table for the left-hand side x (y z)
y x y z x 0 0 0 0 1 1 0 0 1 0 1 1 1 z Truth table for the left-hand side x (y z)
y z x (y z) x y z x 0 0 0 0 1 0 0 0 0 0 1 1 0 1 0 0 1 1 1 Truth table for the left-hand side
x y z (x y) z 0 0 0 0 1 0 0 0 1 1 0 1 0 0 0 1 0 1 1 0 0 1 1 1 Truth table for the right-hand side
These two are identical, which concludes the proof.
x y z 0 0 0 1 1 1 0 0 1 1 1 x y+ z Truth table for the left-hand side x+(y+z)
y+ x y z x 0 0 0 0 1 0 1 0 1 1 0 0 1 0 1 1 1 1 z Truth table for the left-hand side x+(y+z)
y+ z x y z x 0 0 0 0 1 0 1 0 0 1 1 0 1 1 1 1 1 1 1 Truth table for the left-hand side x+(y+z)
x y z x+y z (x+y)+z 0 0 0 0 1 0 1 1 1 0 0 1 1 1 0 1 1 1 1 Truth table for the right-hand side
These two are identical, which concludes the proof.
The Venn Diagram Representation x y z x y +z
Venn Diagram Basics (a) Constant 1 (b) Constant 0 x x x (c) Variable x x (d) x [ Figure 2. 14 from the textbook ]
Venn Diagram Basics x y (e) x y (f) x + y x x y y z (g) x y (h) xy +z [ Figure 2. 14 from the textbook ]
Let’s Prove the Distributive Properties
x y z z (a) x (d) x y x y z z (b) y + z (e) x z x y z z (c) x ( y + z) (f) x y + x z [ Figure 2. 15 from the textbook ]
[ Figure 2. 17 from the textbook ]
Try to prove these ones at home
De. Morgan’s Theorem
Proof of De. Morgan’s theorem [ Figure 2. 13 from the textbook ]
Alternative Proof of De. Morgan’s theorem [ Figure 2. 18 from the textbook ]
Let’s prove De. Morgan’s theorem
Try to prove these ones at home
Venn Diagram Example Proof of Property 17 a
Left-Hand Side x y x y z z z xy xz yz x y + x z + yz [ Figure 2. 16 from the textbook ]
Left-Hand Side x x y y x y z z z xy xz yz x y + x z + yz Right-Hand Side x y x y z z z x y x z x y+ x z [ Figure 2. 16 from the textbook ]
Left-Hand Side x x y These two are equal y x y z z z xy xz yz x y + x z + yz Right-Hand Side x y x y z z z x y x z x y+ x z [ Figure 2. 16 from the textbook ]
Questions?
THE END
- Stoytchev 281
- Stoytchev 281
- Cpr e 281
- Stoytchev
- Stoytchev
- Ipx 281
- Cs281 wordpress
- It is a circular or elliptical anticlinal structure
- Granite weathering
- 281-284-0027
- Cs 281
- Numeros romanos del 1 al 100
- Dönem ayırıcı hesaplar örnek
- Dönem ayirici hesaplar 180 181 280 281 380 381
- First order logic vs propositional logic
- First order logic vs propositional logic
- Third order logic
- Combinational vs sequential logic
- Tw
- Software development plan
- Combinational logic sequential logic
- Combinational logic sequential logic 차이
- Logic chapter three
- Digital logic design tutorial
- Sequential machine examples
- Timing diagram of nand gate
- Digital system design
- Plc mixer process control problem
- Canonical form digital logic
- Digital logic design practice problems
- Digital logic
- Digital logic design lectures
- Digital logic structures
- Consensus theorem in digital electronics
- Bubble matching digital logic
- Digital logic and computer architecture
- Uncommon logic digital
- Bubble matching digital logic
- Law of duality in discrete mathematics
- Digital logic design
- Bubble matching digital logic
- Tipos de participantes en cursos de capacitación
- Basic instructor course texas
- Basic instructor course texas
- Basic instructor course tcole
- Pepperball launcher nomenclature
- Subject verb agreement exercise
- Instructor vs teacher
- Cisco certified trainer
- Mptc firearms instructor
- Basic instructor course texas
- Basic instructor course #1014
- Virtual art instructor
- Nfpa 1403
- Human factors instructor
- Instructor operating station
- Catia instructor
- Instructor
- Ac 61-98 plan of action
- Tcole 1014 basic instructor course
- Marksmanship instructor
- How to become an nrp instructor mentor
- Utp cable
- Cbrf training registry
- Nra certified instructor logo
- Naismith was an instructor of
- Please clean your own room
- Tcole advanced instructor course
- Tcole advanced instructor course
- Jrotc marksmanship instructor course online
- Which line is longer illusion
- Acr medical term
- Basic instructor course #1014
- Basic instructor course #1014
- Delmar cengage learning instructor resources
- Instructor office hours
- Extracorporeal cpr
- 3 prinsip sebelum cpr kecuali
- Perbedaan cpr dan mpr
- Bls algorithm
- Objective of cpr
- Cpr objectives
- Seconday survey
- What is llf in first aid
- For adult
- Emotional cpr pdf
- Cpr hand placement sternum
- What are the three c's of cpr
- Unresponsive child cpr