Proving Formula for nAUB Video Lecture | Additional Study Material for Class 11

what you see on your screen is a
relation this relation is for two
non-empty sets a and B n a union B
equals na plus NB minus n a intersection
B now what we are going to do here is to
prove this relation so let's use Venn
diagrams there you see sets a and B this
region represents set a and this region
represents set B now look at this region
this represents a intersection B what
about a minus B this is this region and
B minus a that is this region so let me
show you these three regions once more
there now you can see from the Venn
diagram that they are disjoint sets that
is they do not have any element in
common and when we combine these three
sets they represent a union B there this
region is a union B right so we can say
that n a union B equals n a minus B plus
n a intersection B plus NB minus a now
go back to the Venn diagram look at a
minus V we can say that n a minus B
equals n a minus n a intersection B and
we substitute this in the expression
similarly we can say that n B minus a
that equals n B minus n a intersection B
so we write that as well n a
intersection B and in a intersection B
cancel out and what we are left with
this n a union B equals na plus NB minus
n a intersection B and this proves the
formula in fact let me also give you a
special case of this formula what
happens when a intersection B is 5 that
is sets a and B do not have any common
elements they are disjoint sets and they
look like this in this case n a
intersection B is 0 so we substitute the
value of n a intersection B in the
formula and what do we get n a union B
equals na plus NB minus 0 or n a union B
equals na plus NB so for two disjoint
sets a and B we have n a Union
equals na plus NB so this is the special
case of the formula and please remember
this