Digital Logic - Boolean Algebra, by Yau Kai Shi

BOOLEAN ALGEBRA

                                                 Introduction

•  Boolean Algebra (developed by George Boole) is an algebra for the manipulation of objects that can take on only two logical value,typically true and false.

•  Boolean Variables that only have the two value ‘0’ (false) and ‘1’ (true).

•  Boolean Functions

Logic Functions	Symbols
OR	+
AND	&
NOT	-or ’

-each of the logic function are represented by symbols as described above.

·         Boolean Theorems is a set of identities and laws.

Boolean Identities

OR Function	AND Function	NOT Function
0+0=0	0&0=0	0’=1
0+1=1	0&1=0	1’=0
1+0=1	1&0=0	A’=A
1+1=1	1&1=1
A+0=A	A&0=0
0+A=A	0&A=0
A+1=1	A&1=A
1+A=1	1&A=A
A+A=A	A&A=A

***notes

·         OR (+)

·         AND (&)

·         NOT( ^- or ’)

Boolean Laws

Identity Law	i.            A+0=0      ii.            A& 1 = A
Zero and One Laws	i.            A+1=1      ii.            A&0=0
Idempotent Laws	i.            A + A = A      ii.            A&A= A
Inverse Laws	i.            A+A’=1      ii.            A&A’ = 1
Commutative Laws	i.            A + B = B + A      ii.            A&B = B&A
Associative Laws	i.            A + (B+C) = (A+B) + C      ii.            A& (B&C) = (A&B) &C
Distributive Laws	i.            A + (B&C) = (A+B) & (A+C)      ii.            A& (B&C) = (A&B) + (A&C)
Absorption Laws	i.            A + A&B  = A    / A + A’B = A+B      ii.            A (A + B) =A

            







symbol i. is OR form

symbol ii. is AND form

Examples of Boolean Algebras

Example 1:

A’B + A’B’ + AC + AC’ = A’(B+B’) + A(C+C’)

                      = A’ (1) + A (1)

          = A +A’

   = 1

Example 2:

xyz + xz’ + xy’z = xyz + xy’z + xz’

                         =xz(y + y’) + xz’

                         = xz + xz’

                         = x (z +z’)

                         = x

                                                

Example 4:

X(W’Z + WZ)+ XY = XZ(W’+ W)+XY

                               =XZ + XY

                               =X(Z+Y)

References:

1.  http://home.adelphi.edu/~siegfried/cs371/371l3.pdf
2.   http://courses.cs.vt.edu/cs2505/fall2012/Notes/T32_BooleanAlgebra.pdf
3.   http://books.google.com.my/books?id=f83XxoBC_8MC&pg=PA122&lpg=PA122&dq=boolean+algebra+computer+organization+architecture&source=bl&ots=5ekB1eR8V9&sig=rAwtSI4K3n6Il26PjUQiIxfX6t8&hl=en&sa=X&ei=QFBtUNuzLorirAfZk4GwBQ&ved=0CCMQ6AEwAQ#v=onepage&q=boolean%20algebra%20computer%20organization%20architecture&f=false
4.  http://www.ee.surrey.ac.uk/Projects/Labview/boolalgebra/index.html
5.  http://www.google.com.my/imgres?q=Boolean+Algebra&hl=en&sa=X&biw=1069&bih=599&tbm=isch&prmd=imvnsb&tbnid=coGALtyulgW5BM:&imgrefurl=http://www.tpub.com/neets/book13/54h.htm&docid=t8ID4Z21BOIsdM&imgurl=http://www.tpub.com/neets/book13/NF130220.GIF&w=900&h=751&ei=hVFtUOHCDYbMrQf8lICYBA&zoom=1&iact=rc&dur=0&sig=116109911643460835037&page=1&tbnh=123&tbnw=147&start=0&ndsp=17&ved=1t:429,r:2,s:0,i:76&tx=599&ty=23

Written by,

Yau Kai Shi

B031210077