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#
