Please enable JavaScript.
Coggle requires JavaScript to display documents.
Simplifying Boolean Algebra (Rule 6: Absorption (To be able to apply the…
Simplifying Boolean Algebra
Rule 1: De Morgans Law
Either logical functions AND or OR may be replaced by the other, given certain changes to the equation
¬(A^B)=(¬A)v(¬B) NOT(A AND B) is the same as (NOT A) OR (NOT B)
¬(AvB(¬A)^(¬B) NOT (A OR B) is the same as (NOT A)AND(NOT B)
Rule 2: Distribution
This law allows for the multiplying or factoring out of an expression
This is the OR Distribution law: A^(BvC) = (A^B)v(A^C) A AND (B OR C) is the same as (A AND B ) OR (A AND C)
This is the AND Distribution law: Av(B^C) = (AvB)^(AvC) A OR (B AND C) is the same as (A OR B) AND (A OR C)
Rule 3: Association
This law allows for the removal of brackets from an expression and the regrouping of the variables
This is the OR Association Law: Av(BvC) = (AvB)vC= AvBvC A OR (B OR C) is the same as (A OR B ) OR C is the same as A OR B OR C
This is the AND Association Law: A^(B^C) = (A^B)^C = A^B^C A AND (B AND C) is the same as (A AND B) AND C is the same as A AND B AND C
Rule 4: Commutation
The order of application of two separate terms is not important
A^B = B^A The order in which two variables are AND'ed makes no difference
AvB = BvA The order in which two variables are OR'ed makes no difference
Rule 5: Double Negation
If you reverse something twice you end up back where you started
¬(¬A) = A
Rule 6: Absorption
The second term inside the bracket can always be eliminated and "Absorbed" by the term outside the bracket of the given rules are met
Xv(X^Y)=X XOR(X AND Y) is the same as X
X^(XvY) = X X AND (X OR Y) is the same as X
To be able to apply the Absorption rule:
The operator inside and outside the brackets must be different.
The term outside the brackets must also be inside the brackets.
General Rules
In addition to the 6 laws already covered there are also 8 general identities or "Rules" which will help when it comes to simplifying Boolean Expressions
These Fall into two categories:
AND rules: For simplifying AND expressions
X^0=0
X AND 0 is the same as 0 Or to put it another way X AND FALSE has to equal FALSE
X^1+X
X AND 1 is the same as X Or to put it another way X AND TRUE has to be equal to TRUE
X^X=X
X AND X is the same as X Or to put it another way X AND X has to be equal to X
X^¬X=0
X AND not X is the same as 0 Or to put it another way X AND NOT(X) has to be equal to False
OR rules: For simplifying OR expressions
Xv0=X
X OR 0 is the same as X Or to put it another way X OR FALSE has to be equal to TRUE
Xv1=1
X OR 1 is the same as 1 Or to put it another way X OR TRUE has to equal TRUE
XvX=X
X OR X is the same as X Or to put it another way X OR X has to equal X
Xv¬X=1
X OR not X is the same as 1 Or to put it another way X OR NOT(X) has to be equal to TRUE