Dual statement
The dual of any statement in a Boolean Algebra is the statement obtained by interchanging the operations `+` and `*` and simultaneously interchanging the elements 0 and 1 in the original statement.
|
Statement |
Dual Statement |
|
1. `x*y` |
`x+y` |
|
2. `x*(y+z)=(x*y)+(x*z)`
|
`x+(y*z)=(x+y)*(x+z)`
|
|
3. `x+0=x`
|
`x*1=x`
|
1. Prove that `x+x=x`
LHS `=x+x`
`=(x+x)*(1)`
`=(x+x)*(x+x')`
`=x+(x*x')`
`=x+0`
`=x`
`=`RHS
2. Prove that `x*x=x`
LHS `=x*x`
`=(x*x)+(0)`
`=(x*x)+(x*x')`
`=x*(x+x')`
`=x*1`
`=x`
`=`RHS
3. Prove that `x+1=1`
LHS `=x+1`
`=(x+1)*(1)`
`=(x+1)*(x+x')`
`=x+(1*x')`
`=x+x'`
`=1`
`=`RHS
4. Prove that `x*0=0`
LHS `=x*0`
`=(x*0)+(0)`
`=(x*0)+(x*x')`
`=x*(0+x')`
`=x*x'`
`=0`
`=`RHS
5. Prove that `x+(x*y)=x`
LHS `=x+(x*y)`
`=(x*1)+(x*y)`
`=x*(1+y)`
`=x*1`
`=x`
`=`RHS
6. Prove that `x*(x+y)=x`
LHS `=x*(x+y)`
`=(x+0)*(x+y)`
`=x+(0*y)`
`=x+0`
`=x`
`=`RHS
This material is intended as a summary. Use your textbook for detail explanation.
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