Question 1123415
if n(A)=80,n(B)=30,n(A U B)=100 then find n [(A-B) U (B-A)]<pre>
{{{drawing(360,240,-4,4,-2,4.8,

locate(-2,1.8,x),locate(1.5,1.7,z),



 locate(-3.6,2.5,A), locate(-.1,1.8,y),

red(circle(-sqrt(2),sqrt(2),2)),

red(circle(-sqrt(2),sqrt(2),1.95)),

red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),


locate(3.4,2.5,B)
 )}}}

n(A) = x+y
n(B) = y+z
n(A-B) = (x+y)-y = x, 
n(B-A) = (y+z)-y = z,
n[(A-B) U (B-A)] = x+z since (A-B) and (B-A) are disjoint

n(A) = x+y = 80
n(B) = y+z = 30
n(A U B) = x+y+z = 100 

So we have the system of three equations in three unknowns:

x + y     =  80
    y + z =  30
x + y + z = 100

Subtracting the first equation from the third gives z = 20
Substituting that in the second equation gives y = 10
Substituting that in the first equation gives x = 70

Now our Venn diagram becomes

{{{drawing(360,240,-4,4,-2,4.8,

locate(-2,1.8,x=70),locate(1.5,1.7,z=20),

locate(-3.6,2.5,A), locate(-.4,1.8,y=10),

red(circle(-sqrt(2),sqrt(2),2)),

red(circle(-sqrt(2),sqrt(2),1.95)),

red(circle(-sqrt(2),sqrt(2),1.975)),
blue(circle(sqrt(2),sqrt(2),2),circle(sqrt(2),sqrt(2),1.95),circle(sqrt(2),sqrt(2),1.975)),


locate(3.4,2.5,B)
 )}}}


n[(A-B) U (B-A)] = x+z = 70+20 = 90

Edwin</pre>