Boolean Algebra Proofs Postulates and Theorems (Part 1)

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.

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