Question 1055232
.
n(A∩B) = n(A) + n(B) - n(A∪B) = 13 + 15 - 19 = 9.


<U>Proof</U>


It is easier to prove an equivalent equality


n(A&#8746;B) = n(A) + n(B) - n(A&#8745;B).


Let us try to count all elements in A&#8746;B.

As a first approximation, we will take  n(A) + n(B).
But doing so, we count the elements of the intersection  n(A&#8745;B)  twice.
So, we need distract  n(A&#8745;B)  from  n(A) + n(B), and in this way we get the exact number of elements in  A&#8746;B.


The proof is completed.



It is a classic problem of elementary set theory.


See also the lesson 

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=http://www.algebra.com/algebra/homework/word/misc/Counting-elements-in-sub-sets-of-a-given-finite-set.lesson>Counting elements in sub-sets of a given finite set</A>

in this site.



Also, you have this free of charge online textbook in ALGEBRA-I in this site

&nbsp;&nbsp;&nbsp;&nbsp;- <A HREF=https://www.algebra.com/algebra/homework/quadratic/lessons/ALGEBRA-I-YOUR-ONLINE-TEXTBOOK.lesson>ALGEBRA-I - YOUR ONLINE TEXTBOOK</A>.


The referred lesson is the part of this online textbook under the topic "<U>Miscellaneous word problems</U>".