Hi all, we have been learning some Boolean Algebra in my Computer Architecture class and I thought I would share what I have learned so far. As always, if you have any questions or comments please share them below.
Boolean Algebra is the subset of Algebra formulated by English mathematician George Boole. In "normal Algebra" the variables are numbers and the main operations are addition, subtraction, multiplication and division. But, in Boolean Algebra the values of the variables are true and false (1 and 0) and the main operations are "and" (^), "or" (v) and "not". Now, why is this significant? How is Boolean Algebra useful and why couldn't we just use Algebra? Well, the laws of Boolean Algebra are mainly used to describe circuits whose state can either be true or false (1 or 0). Boolean Algebra differs from Algebra because the values of true and false are represented by binary digits (bits) that do not behave like normal integers (think about the NOT, AND, and OR gates in my previous post). Don't get confused, in Boolean Algebra the 1 and 0 do NOT behave like normal integers, they represent true and false.
Boolean Algebra is the subset of Algebra formulated by English mathematician George Boole. In "normal Algebra" the variables are numbers and the main operations are addition, subtraction, multiplication and division. But, in Boolean Algebra the values of the variables are true and false (1 and 0) and the main operations are "and" (^), "or" (v) and "not". Now, why is this significant? How is Boolean Algebra useful and why couldn't we just use Algebra? Well, the laws of Boolean Algebra are mainly used to describe circuits whose state can either be true or false (1 or 0). Boolean Algebra differs from Algebra because the values of true and false are represented by binary digits (bits) that do not behave like normal integers (think about the NOT, AND, and OR gates in my previous post). Don't get confused, in Boolean Algebra the 1 and 0 do NOT behave like normal integers, they represent true and false.
Boolean Postulates:
- 0 . 0 = 0
- 0 + 0 = 0
- 1 . 1 = 1
- 1 + 1 = 1
- 1 . 0 = 0 . 1 = 0
- 0 . 1 = 1 . 0 = 1
Boolean Laws:
AND ( . )
OR ( + )
Every law has a part (a) and a part (b) this is called duality and is done by switching every AND for an OR and every 1 for a 0 (inverting it).
OR ( + )
Every law has a part (a) and a part (b) this is called duality and is done by switching every AND for an OR and every 1 for a 0 (inverting it).
- Commutative Law
- (a) A + B = B + A
- (b) A . B = B . A
- Associate Law
- (a) (A + B) + C = A + (B +C)
- (b) ( A . B ) . C = A + (B . C)
- Distributive Law
- (a) A (B + C) = A . B + A. C
- (b) A + (B . C) = (A + B) . (A +C)
- Identity Law
- (a) A + A = A
- (b) A . A = A
- Redundance Law
- (a) A + A . B = A
- (b) A . (A + B) = A
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