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4.2.4
Just like "normal" algebra and equally useful. When parentheses (brackets) don't matter, they don't matter.
Somewhat similar to "multiplying out". Makes sense to me (try a truth table to check!) 1 = true; 0 = false; You can see this works by applying 3 then 4.
(NOT the individual terms. Change the sign. NOT the lot).
We often look for patterns that will allow the production of terms such as ( not x + x) which can then be eliminated. This is a bit tedious and can be error prone.
Many students prefer to simplify via Karnaugh maps as it is a more visual approach.
A  sensor to detect lights left on. B  Sensor to tell if car is left in gear C  Sensor to detect of keys are still in ignition P  the buzzer (alarm)
The truth table indicates all the possible combinations of inputs. Notice it follows a set sequence  counts up in binary. We are only interested in the cases where the output is a binary 1.
tautology (a)
Paired up 1 and 3, 2 and 4.
Eliminate terms in brackets using tautology (b) Showing that this expression simplifies to A.
Next  an easier way? Karnaugh Maps

On this page: [ expressions  using algebraic laws ] Simplifying Expressions The first and (for some students, most difficult) method of minimization is by use of algebra. Boolean algebra has 8 laws : 1. Commutative laws A + B = B + A A . B = B . A 2. Associative laws A + (B + C) = (A + B) + C A . (B . C) = (A . B) . C 3. Distributive laws A . (B + C) = (A . B) + (A . C) A + (B . C) = (A + B) . (A + C) 4. Tautology laws (a) A.A = A A + A = A (b) A + NOT A = 1 A . NOT A = 0 5. Absorption laws A + (A . B) = A A . (A + B) = A 6. Common sense laws (a) 0 . A = 0 1 + A = 1 (b) 1 . A = A 0 + A = A (c) NOT 0 = 1 NOT 1 = 0 7. Double complement law NOT NOT A = A 8. De Morgan's law (NOT A + NOT B) = NOT (A . B) (NOT A . NOT B) = NOT (A + B) Using the laws Given an expression, such as: We can simplify it using the above laws. using distributive law: not X( not Y.Z + Y. not Z) + X( not Y.Z + Y. not Z) associative: ( not X + X).( not Y.Z + Y. not Z) gives not Y.Z + Y not Z In the context of IB problems on this topic, you will often be presented with a threeinput device. For example: Consider my car which complains by sounding a buzzer when I have left the lights on or left the car in gear (not in Park) and taken the keys out of the ignition: To simplify using algebraic laws we attempt to identify patterns which will eliminate terms. If I look at the first and third terms, for example, I see that it contains opposite A terms but that the B and not C are the same. Therefore I could eliminate A from this pair. Similarly B can be eliminated from the second and third terms. However, either of these will leave me with a clumsy looking equation. Since I can see how to use the third term twice it can simply be aded again using the reverse of the tautology law: A + A = A giving now we can pair up: and eliminate: One more example for good measure: related: [ Topic 4 home  previous: truth tables  next: Karnaugh maps ] 
is also known as "minimization" since we are trying to reduce the number of variables (and therefore gates in a circuit). 


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