12. (i) Prove that the sum of all minterms of a Boolean function of 3 variables is 1.
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Hello we know
if x,y,z are Boolean variable then
x+y +z = 1 and x.y.z = 0
consider
x = 1 , y = 0 , z = 0
we know 1+0+0 = 1 , 0×1 ×0= 0
=> x + y +z= 1 , x.y.z =0
let x = 1, y = 1 , z=0
=>x + y +z = 1 , x.y.z =0
since 1+1+0 =1 , 1×1×0 =0
now let x=1, y=1,z=1
=> x+y+z = 1 , x.y.z =1
since 1+1+1 = 1 , 1×1×1 = 1
similarly you can prove each case .
I have also shown you the product also .
I hope it will help you ✌
if x,y,z are Boolean variable then
x+y +z = 1 and x.y.z = 0
consider
x = 1 , y = 0 , z = 0
we know 1+0+0 = 1 , 0×1 ×0= 0
=> x + y +z= 1 , x.y.z =0
let x = 1, y = 1 , z=0
=>x + y +z = 1 , x.y.z =0
since 1+1+0 =1 , 1×1×0 =0
now let x=1, y=1,z=1
=> x+y+z = 1 , x.y.z =1
since 1+1+1 = 1 , 1×1×1 = 1
similarly you can prove each case .
I have also shown you the product also .
I hope it will help you ✌
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