idempotent law in a boolean algebra

show that a + a = a in a boolean algebra

a = a + 0
= a + (a. a' )
= (a + a ). (a + a ' )
= (a + a ). 1
= (a + a )

therefore a + a = a
note that the dual is a . a = a

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boundedness law in a boolean algebra

Show that a +1 = 1 in a boolean algebra

1 = a + a'
= a + (a' .1 )
= (a + a' ) . ( a+1) distributive law
= 1 . ( a+1)
=a +1

therefore a + 1 = 1

note that the dual is a . 0 = 0

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to construct truth tables for [~ (p ^ q ) ]v (~r)

since there are three variables there will be 2^3 =8 rows
start the first column with 4 T's and 4F's
second column with 2 T's and 2F's ...
third column T and F alternatively
then use the definition of ^ , ~ to get the truth table of the components and finally that of the question
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variation of parameter method

solve xy" - 4y' = x^4 by method of variation of parameter
make the coefficient of y" unity by dividing by x

find CF in the form Ay1 + By2 (using trial and error or some other method)
find the wronskian
and then find the PI as shown below

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If n is an integer, show that one of the integers n, n+2, or n+4 is divisible by 3.

If n is an integer, show that one of the integers n, n+2, or n+4 is divisible by 3.

solution:
If n is a multiple of 3, then there is nothing more to prove.

Suppose n is not a multiple of 3, then n must be of the form
n =3k+1 or n = 3k+2, for some integer k

if n=3k+1,
then n+2 = (3k+1)+2 = 3k+3 = 3(k+1) = multiple of 3

if n = 3k+2,
then n+4 = 3k+6 = 3(k+2) =multiple of 3


hence the result

some other math problems

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