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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.
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?
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)
( 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 three-input 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
now we can pair up:
One more example for good measure:
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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This page was last modified: October 28, 2013