Question #bcf5c

1 Answer
Dec 2, 2017

Please refer to the explanation.

Explanation:

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#