Question

a)If n(A)=4 and n(B)=10,find: 1. max n(A ∪ B) 2. min n(A ∪ B) 3. max n( A ∩ B ) 4. min n(A ∩ B ) .​

Answer 1

Answer:

Step-by-step explanation:

Let’s take a look at a couple of Venn diagrams. First, defining the regions:

A′  means “not in  A ”. So you can see that region  A  is made up of two parts. One part has some of  B  included. The other part does not.

The middle part,  A∩B is part of  A  and part of  B  at the same time. Since we know  n(A)=21 (the number in  A ) and  n(B)=43 , we can parse out the regions like this.

We know that  A  has to add to 21 and  B  has to add to 43. But there is a potential shared region, the Intersection.

If you add  A  and  B  together, you will get  64–x . But there are only 60 in the Universe, so  60–(64–x)=x–4 .

Be sure to check. Add all the regions and confirm that there are a total of 60.

Now, what possibilities do we have?  x  has to be at least 4. You can’t have negative quantities of objects in a set. If you let  x=4 , then  A∪B=60 . That means that the outside has 0 elements in it. All elements are contained in the union of  A  and  B .

But  x  cannot be more than 21. That would mean that every element in  A  is also in  B , so the minimum for  A∪B=43