Hello Readers.
I will begin the journey to Boolean enlightenment with the fundamental postulates of Boolean algebra.
For all the following, the variables (a,b,c...) will have two possible values of either 1 (true) or 0 (false) represented as such : a = 1 , a' = 0.
• Basic Postulates:
• Postulate 1 (Definition): A Boolean algebra is a closed algebraic system
containing a set K of two or more elements and the two operators . and +.
• Postulate 2 (Existence of 1 and 0 element):
(a) a + 0 = a (identity for +), (b) a .1 = a (identity for .)
• Postulate 3 (Commutativity):
(a) a + b = b + a, (b) a .b = b .a
• Postulate 4 (Associativity):
(a) a + (b + c) = (a + b) + c (b) a. (b.c) = (a.b) .c
• Postulate 5 (Distributivity):
(a) a + (b.c) = (a + b) .(a + c) (b) a. (b + c) = a.b + a.c
• Postulate 6 (Existence of Complement):
(a) a + a' = 1 (b) a.a' = 0
Usually the '.' sign is ommited.
FUNDAMENTAL THEOREMS OF BOOLEAN ALGEBRA:
- Theorem 1 (Idempotency): (a) a + a = a (b) aa = a
- Theorem 2 (Null Element): (a) a + 1 = 1 (b) a0 = 0
- Theorem 3 (Involution): a'' = a
- Theorem 4 (Absorption): (a) a + ab = a (b) a (a + b) = a
- Theorem 5: (a) a + a'b = a+b (b) a (a' + b) = ab
- Theorem 6: (a) ab + ab' = a (b) (a + b)(a + b') = a
- Theorem 7: (a) ab + ab'c = ab + ac (b) (a + b)(a + b' + c)= (a + b)(a + c)
- Theorem 8 (Demorgan's Theorem): (a)(a + b)' = a'b' (b) (ab)' = a' + b'
- Theorem 9(Consensus): (a) ab + a'c + bc = ab + a'c (b) (a + b)(a' + c)(b + c)= (a + b)(a + c)
This is all pretty basic logic (1+0 = 1 and 1.0 = 0), applicable to general algebraic thinking. If 1 is substituted for all the normal letters and 0 for all the 'complemented' letters (a' , b' , c'), you will find that these theorems are in fact very simple. They are powerful tools for simplifying very seemingly complex functions and circuits and any aspiring EECE student would be wise to memorize these.
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