Boolean Algebra Postulates and Theorems (Part 1):
First familiarize with truth tables so it’ll be easier to understand.
x + 0 = x
here only two possible states of x, 0 remains constant
x = 0 x = 1
So,
false OR false is always false 0 + 0 = 0 true OR false is always true 1 + 0 = 1
So, from this we can see whatever the value of x is, the output is always equal to x.
x . 1 = x
here only two possible states of x, 1 remains constant
x = 0 x = 1
So,
false AND true is always false 0 . 1 = 0 true AND true is always true 1 . 1 = 1
So, from this we can see whatever the value of x is, the output is always equal to x.
x + 1 = 1
here only two possible states of x, 1 remains constant
x = 0 x = 1
So,
false OR true is always true 0 + 1 = 1 true OR true is always true 1 + 1 = 1
So, from this we can see no matter the value of x, OR with 1 (true) always gives a 1 (true) value.
x . 0 = 0
here only two possible states of x, 0 remains constant
x = 0 x = 1
So,
false AND false is always false 0 . 0 = 0 true AND false is always false 1 . 0 = 0
So, from this we can see no matter the value of x, AND with 0 (false) always gives a 0 (false) value.
x + x’ = 1
x = 0, x' = 1 x = 1, x' = 0
So,
false OR true is always true 0 + 1 = 1 true OR false is always true 1 + 0 = 1
x . x’ = 0
x = 0, x' = 1 x = 1, x' = 0
So,
false AND true is always false 0 . 1 = 0 true AND false is always false 1 . 0 = 0
x + x = x
x = 0, x = 0 x = 1, x = 1
So,
false OR false is always false 0 . 0 = 0 true OR true is always true 1 . 1 = 1
So, we can whatever the value of x is, that is our output.
x . x = x
x = 0, x = 0 x = 1, x = 1
So,
false AND false is always false 0 . 0 = 0 true AND true is always true 1 . 1 = 1
So, again we can whatever the value of x is, that is our output.
(x’)’ = x
x = 0, x' = 1, (x')' = 0 x = 1, x' = 0, (x')' = 1
So, we can see complementing twice gives the original value.
