A-level Computing 2009/AQA/Computer Components, The Stored Program Concept and the Internet/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)

Examples of manipulating and simplifying simple Boolean expressions.

Let's try to simplify the following: $$A+B+B$$ Using the rule $$B + B = B$$ $$A+B+B = A+B$$ Trying a slightly more complicated example: $$(A.0)+B$$ dealing with the bracket first $$(0)+B$$ as $$0.A = 0$$ $$B$$ as $$0+B = B$$ $$(A.0)+B = B$$

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:

$$(A.B) + A$$ Where can we go from here, let's take a look at some identities Now for something that requires some 'multiplication'
 * 1) $$(A.B) + (A.1)$$ using the identity A = A.1
 * 2) $$A.(B+1)$$ taking the common denominator from both sides
 * 3) $$A.1$$ as B+1 = 1
 * 4) $$A$$
 * 1) $$(\overline{A}.B) + A$$
 * 2) $$(\overline{A}+A).(B+A)$$multiply it out
 * 3) $$1.(B+A)$$cancel out the left hand side as  $$(\overline{A}+A)=1$$
 * 4) $$B+A$$using the identity $$1.Q = Q$$