We must prove, using the properties of sets, that
[ ( A' uu B' ) - A ] ' = A .... "" color(blue)(Equation.1)
color(red)(Step.1)
De Morgan's Law states that
( A' uu B' ) hArr (A nn B)'
Hence we can rewrite "" color(blue)(Equation.1) as
[ (A nn B)' - A ] ' = A .... "" color(blue)(Equation.2
color(red)(Step.2)
Set Difference Law states that
{A - B} rArr {x:x in A and x !in B}.... color(green)(Property)
That is, we include all elements in Set A that is NOT in Set B .
color(red)(Step.3)
We will now rewrite [ (A nn B)' - A ] from "" color(blue)(Equation.2 using color(green)(Property) above as
[ (A nn B)' - A ] rArr {x:x in (A nn B)' and x !in A} ... which refers to Set A.
Now it is obvious that
[ (A nn B)' - A ]' rArr A'
Please note that [A']' = A
color(red)(Step.4)
Hence, we have proved that
[ ( A' uu B' ) - A ] ' = A
