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Boolean Algebra, Logic Diagrams and Truth Tables: Kjartan Halvorsen

This document covers Boolean logic concepts. It defines AND, OR, NAND, NOR logic gates using truth tables and shows how logic diagrams visually represent Boolean functions. The document also discusses properties of Boolean algebra including idempotency, commutativity

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Bryan Badillo Ramos
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76 views10 pages

Boolean Algebra, Logic Diagrams and Truth Tables: Kjartan Halvorsen

This document covers Boolean logic concepts. It defines AND, OR, NAND, NOR logic gates using truth tables and shows how logic diagrams visually represent Boolean functions. The document also discusses properties of Boolean algebra including idempotency, commutativity

Uploaded by

Bryan Badillo Ramos
Copyright
© © All Rights Reserved
We take content rights seriously. If you suspect this is your content, claim it here.
Available Formats
Download as PDF, TXT or read online on Scribd
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Boolean algebra, logic diagrams and truth tables

Kjartan Halvorsen

April 15, 2020


AND and OR

a, b ∈ {0, 1}

AND OR
a b a AND b, ab a b a OR b, a + b
0 0 0 0 0 0
0 1 0 0 1 1
1 0 0 1 0 1
1 1 1 1 1 1
a b a

Closed circuit ⇔ 1
Open circuit ⇔ 0
b
a
ab a
a+b
b b
NAND and NOR

a, b ∈ {0, 1}

NAND NOR
a b a NAND b, a · b a b a NOR b, a + b
0 0 1 0 0
0 1 1 0 1
1 0 1 1 0
1 1 0 1 1
a a
a·b a+b
b b
Boolean algebra, contd

x, y , z ∈ {0, 1}
Property Dual
Properties of 0 and 1 x +0=x x ·0 = 0
x +1=1 x ·1 = x
Idempotency x +x =x x ·x = x
Complementarity x +x =1 x ·x = 0
Involution x =x
Commutative x +y =y +x x ·y = y ·x
Associative (x + y ) + z = x + (y + z) (xy )z = z(yz)
Distributive x · (y + z) = xy + xz x + yz = (x + y )(x + z)
Boolean algebra, contd

x, y ∈ {0, 1}
Theorem Dual
Absorption x + xy = x(1 + y ) = x x(x + y ) = x
Logic adjacency xy + xy = x(y + y ) = x (x + y )(x + y ) = x
De Morgan’s x + y = x ·y xy = x + y
DeMorgan’s theorem

From wikipedia
Simplify functions

1. f = (a + b)(a + c)
2. f = a + ab
Logic diagram → function
Determine the function represented by the logic diagrams
a a
b b
f f
c
a
c

a a
b b
f f
b̄
d
c
Function → logic diagram

Draw the diagram corresponding to the boolean function


1. f = (a + b)(a + c)
2. f = a + ab
Group exercise

1. Enter breakout room


2. One of you downloads and shares this presentation
3. Work together on the problems in the previous three slides
3.1 Simplify functions
3.2 Determine function from logic diagram
3.3 Draw logic diagram from function

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