probability,(If E C F, then F^c C E^c) Q2 prove that P(EF^c) = P(E)-P(EF)

Question

this a probability question.it is from my school math question

Answer 1

Q1

Suppose E C F. This means every x in E is also in F.  Assume y is not in F.  Then either y is in E or not in E. By E C F, y can't be in E. Otherwise, y in E implies  y in F, which is contrary to our assumption on y. Thus, y is not in E.  The bolded statements say F^c C E^c 

 

Q2

Please note the following fact (which you can convince yourself by drawing a venn diagram): E = E∩F ∪ E∩F^c

 

Because F and F^c are disjoint sets. It follows that E∩F and E∩F^c are disjoint.

Thus, P(E) = P(E∩F ∪ E∩F^c) = P(E∩F) + P(E∩F^c). 

 

Rearrange the above to get, P(E∩F^c) = P(E) - P(E∩F).