Sets (Venn Diagrams) Really need help asap!

Question

There are 35 girls in a class. 18 of these girls are in Dance and 21 girls are in Band. Given that

ξ = {Girls in the class}

D = {Girls who are in Dance}

B = {Girls who are in Band}

(i) Find the smallest value of n(D ∩ B).

(ii) Express the following in set notation: {Girls who is neither in Dance nor Band}

 

I need step by step answers for the whole question as I am unsure how to get the smallest value for part (i). Can't the smallest value for part (i) be 0? 

Answer 1

Theoretically it can be zero, but not in this particular case.  

 

Notice that if you add the number of girls in each activity, there are 39.  Since there are 35 girls in the class, (39-35 = 4) says that at least 4 girls are in both. 

 

It is possible that Mary is in the class but not in either activity.  That would mean that 5 girls are in both.

 

Break  the diagram into groups;   A group of girls in dance, a group of girls in band, a group of girls in both and a group of girls in neither.  Use a few individual names if that helps you see the possibilities.