What is Boolean algebra: Definition and 153 Discussions
In mathematics and mathematical logic, Boolean algebra is the branch of algebra in which the values of the variables are the truth values true and false, usually denoted 1 and 0, respectively. Instead of elementary algebra, where the values of the variables are numbers and the prime operations are addition and multiplication, the main operations of Boolean algebra are the conjunction (and) denoted as ∧, the disjunction (or) denoted as ∨, and the negation (not) denoted as ¬. It is thus a formalism for describing logical operations, in the same way that elementary algebra describes numerical operations.
Boolean algebra was introduced by George Boole in his first book The Mathematical Analysis of Logic (1847), and set forth more fully in his An Investigation of the Laws of Thought (1854).
According to Huntington, the term "Boolean algebra" was first suggested by Sheffer in 1913, although Charles Sanders Peirce gave the title "A Boolean Algebra with One Constant" to the first chapter of his "The Simplest Mathematics" in 1880.
Boolean algebra has been fundamental in the development of digital electronics, and is provided for in all modern programming languages. It is also used in set theory and statistics.
An example in the book asks us to implement the XOR (exclusive-or) function using only 2-input NAND gates.
So:
f = x_1 \overline{x_2} + \overline{x_1}x_2
If we let \uparrow represent the NAND function. That means that: f = (x_1 \uparrow \overline{x_2}) \uparrow (\overline{x_1} \uparrow...
I think I am missing part of my notes, or at least I don't understand them:
if x+y = y+z and xy = xz, then x=z
x = (y+z)x Absorbtion (Don't really know where this is coming from)
x(y+z) Commutative
xy+xz Distributive
It stops here and starts again at:
yz+xz (I have...