ok...

I'll try to start with the basics. (which is probably stuff you already understand).

in regular algebra a letter is a symbol for an unknow number

eg, 1 + x = 2 and you know that x is one.

in boolean algebra, the letters stand for states, that will give a particular logic output. (1 or 0)

there are four symbols

a dot (.) is AND (both)

a plus (+) is OR (either)

a slash (/) is NOT (and in that one, but not that one) (if you were writting it on paper a line over the top is recognised as not). (as a not this is also called a prime)

other than that it works almost exactly the same as normal maths.

for example 2 * 3 = 6 and 3 * 2 = 6 also 6 = 3 * 2

2 + 3 = 5 and 3+2 = 5

in boolean algebra this still works exactly the same

x . y = y . x

x + y = y + x

that's the commutive law.

you use parenthesis to denote the grouping of expression

for example in regular maths

2(3+5) = 16

you're basically grouping the 3+5 and then applying the part that comes outside the brackets.

it works the same way in boolean algebra

a . (b+c)

= a . b + a . c

"A and B or C" is the same as "A and B, or A and C"

you can also use brackets to apply a logic function to what's in the brackets.

A . /(B+C)

A and not B or A and not C

and just like regular algebra

(1 + 2)(3 + 4) = 1 * 3 + 1 * 4 + 2 * 3 + 2 * 4

the same is also true of boolean algebra

(a+b).(c+d) = a.c + a.d + b.c + b.d

that associative law

then there is identity laws, which again work just like normal maths.

a + 0 = a

a . 1 = a

To be honest, I really feel that that is actually all you need to know about boolean algebra in relation to logic states. that enables you to identify black box logic circuits.

there is the complimentaty laws,

A + /A= 1

and A./A = 0

the prime of a prime is zero

/a . /a = 0

and primes cancel each other out

//a = a

(not not A), doub le negative there

then there is demorgans law

if you have the prime of two states /(ab) this is the same as writting /a+/b

Next....

in it's simplest it's easiest to think of this in terms of a light bulb and a set of switches.

lets say I have two switches,

switch a, and switch b

if I turn switch a off the light goes off

if I turn switch a on the light comes on, it doesn't matter what position switch b is in.

to express this in boolean algebra I can

if switch A is on, and switch B is on, the light is on

A.B= 1

or, another set of switch positions that will turn the light on

a = on, b = off

A./B =1

so there are two states that the switches can be in to turn the light on

a and b both on, or a on and b off

A.B + A./B = 1

if you look at this equation it's quite easy to see that a.b or a./b can be reduced just to A=1.

then you see that it's only switch A that does anything.

here's a slightly different explanation of the gates and how to form equations

Boolean arithmetic : BOOLEAN ALGEBRA