Question

show that the logical sum of all minterms of a boolean function of n variables is 1

Answer 1

 I used ' for not and variables a, b, c 
S = abc+ abc'+ ab'c+ ab'c'+ a'bc+ a'bc'+ a'b'c+ a'b'c' = 
ab(c+c') + ab'(c+c')+ a'b(c+c')+ a'b'(c+c')= 
ab+ab'+ a'b+a'b'= 
a(b+b')+ a'(b+b')= 
a+ a' = 1 
We know that x+ x' =1

Answer 2

Step-by-step explanation

Taking the value of N = 3

To prove :

The logical sum of all minterms is 1

S = abc + abc' + ab'c + ab'c' + a'bc + a'bc' + a'b'c + a'b'c'  

= ab ( c + c' ) + ab' ( c + c' ) + a'b ( c + c' ) + a'b'( c + c' )  

We know,

a + a' = 1

Applying this we get,

= ab ( 1 ) + ab' ( 1 ) + a'b ( 1 ) + a'b'( 1 )  

= ab + ab' +  a'b + a'b'  

= a ( b + b' ) + a'( b + b' )  

= a ( 1 ) + a' ( 1 )

= a + a '

= 1

Hence Proved !