
The mathematician George Boole, hence the name Boolean algebra, used 1 for true, 0 for false and + for the or connective to write simpler tables. Let X = "get a salary increase", Y = "win the lottery" and F = "buy a car". The above table can be written in much simpler form as shown below and it defines the OR function.
Let us now examine the following statement:"I will be able to read ebooks online if I buy a computer and get an internet connection." The proposition "read ebooks" depends on two other propositions "buy a computer" and "get an internet connection". Again using 1 for True, 0 for False, F = "read ebooks" , X = "buy a computer", Y = "get an internet connection" and use ^{.} for the connective and, we can write all possible situations using Boolean algebra as shown below. The above table can be written in much simpler form as shown below and it defines the AND function.
We have so far defined two operators: OR written as + and AND written ^{.} . The third operator in Boolean algebra is the NOT operator which inverts the input. Whose table is given below where NOT X is written as X'.
The 3 operators are the basic operators used in Boolean algebra and from which more complicated Boolean expressions may be written. Example: F = X ^{.} (Y + Z)
Rules and Theorems in Boolean AlgebraX + X = XX ^{.} X = X (X')' = X X + X' = 1 X ^{.} X' = 0 X + Y = Y + X , X ^{.} Y = Y ^{.} X : commutative X + (Y + Z) = (X + Y) + Z , X ^{.} (Y ^{.} Z) = (X ^{.} Y) ^{.} Z : associative X ^{.} (Y + Z) = X ^{.} Y + X ^{.} Z DeMorgan's Theorem (x + Y)' = X' ^{.} Y' (x ^{.} Y) = X' + Y'
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