Fundamental Hardware Elements of Computers: Boolean identities
Sometimes a very complex set of gates can be simplified to save on cost and make faster circuits. A quick way to do that is through boolean identities. Boolean identities are quick rules that allow you to simplify boolean expressions. For all situations described below:
A = It is raining upon the British Museum right now (or any other statement that can be true or false) B = I have a cold (or any other statement that can be true or false)
Identity  Explanation  Truth Table  

It is raining AND It is raining is the same as saying It is raining 
 
It is raining AND It isn't raining is impossible at the same time so the statement is always false 
 
2+2=4 OR It is raining. So it doesn't matter whether it's raining or not as 2+2=4 and it is impossible to make the equation false 
 
1+2=4 OR It is raining. So it doesn't matter about the 1+2=4 statement, the only thing that will make the statement true or not is whether it's raining 
 
It is raining OR It is raining is the equivalent of saying It is raining 
 
It is raining OR It isn't raining is always true 
 
1+2=4 AND It is raining. It is impossible to make 1+2=4 so this equation so this equation is always false 
 
2+2=4 AND It is raining. This statement relies totally on whether it is raining or not, so we can ignore the 2+2=4 part 
 
It is raining OR I have a cold, is the same as saying: I have a cold OR It is raining 
 
It is raining AND I have a cold, is the same as saying: I have a cold AND It is raining 
 
It is raining OR (It is raining AND I have a cold). If It is raining then both sides of the equation are true. Or if It is not raining then both sides are false. Therefore everything relies on A and we can replace the whole thing with A. Alternatively we could play with the boolean algebra equation: Using the identity rule 
 
It is raining AND (It is raining OR I have a cold). If It is raining then both sides of the equation are true. Or if It is not raining then both sides are false. Therefore everything relies on A and we can replace the whole thing with A. Alternatively we could play with the boolean algebra equation: Using the identity rule 

Examples of manipulating and simplifying simple Boolean expressions.
Example: Simplifying boolean expressions Let's try to simplify the following:
Using the rule
Trying a slightly more complicated example:
dealing with the bracket first as as 
Exercise: Simplifying boolean expressions
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:
Answer:

Sometimes we'll have to use a combination of boolean identities and 'multiplying' out the equations. This isn't always simple, so be prepared to write truth tables to check your answers:
Example: Simplifying boolean expressions
Where can we go from here, let's take a look at some identities
Now for something that requires some 'multiplication'

Exercise: Simplifying boolean expressions
Answer:
multiplying out
Answer: This takes some 'multiplying' out:
Answer: This takes some 'multiplying' out: treat the brackets first and the AND inside the brackets first multiply it out as as
Answer:
as as take A out as the common denominator as
Answer: This takes some 'multiplying' out:
Answer: This takes some 'multiplying' out: multiplied out as as
Answer: Take the common factor, from both sides: As Then As Then
