Question

Suppose X,Y,Z ⊆ U. If X is the set of all people who played hockey in high school, Y is the set of all out-of-state students, and Z is the set of all international students, describe the following sets in words:

(a) X′ ∪Y ∪Z   (b) X ∩ Y ′ ∩ Z′   (c) (X∩Y′)∪Z′

Consider the set A = {b, c, d}.

(a) How many subsets does A have? (b) List all subsets of A.

Suppose that a committee of 3 people is chosen from 10 people, including Daniel.

(a) How many total committees are there? (b) How many committees include Josh?

(c) Does it seem likely Josh will be chosen for the committee if it is done randomly? Explain.

Answer 1

Answer :-

   Given, X,Y,Z ⊆ U

Where,

X is the set of all people who played hockey in high school

  Y is the set of all out-of-state students

  Z is the set of all international students

Thus,

(a). X′ ∪Y ∪Z

   This event can be expressed in words as follows;

   The set of all people who not played hockey in high school or out-of-state students or international students.

(b). X ∩ Y ′ ∩ Z′

This event can be expressed in words as follows;

   The set of all people who played hockey in high school and who are not out-of-state and national students.

(c). (X∩Y′)∪Z′

This event can be expressed in words as follows;

   The set of all people who played hockey in high school or not out-of-state students and not international students.

If A = {b, c, d} is the given set

then,

(a). How many subsets does A have ?

Any set with n distinct elements has total number of subsets.

   Thus, the given set has total number of subsets.

(b) List all subsets of A.

   Possible subsets are as follows;

   i). A={ } i.e. an empty set

ii). A={ a }    i.e.singleton sets

   iii). A = { b }

   iv). A = { c }

   v). A = { a , b }

vi). A = { a, c }

vii). A = { b , c }

   viii). A = { a, b, c } i.e. each set is a subset of itself.

If a committee of 3 people is chosen from 10 people, including Daniel.

(a) How many total committees are there?

  

(b) How many committees include Josh?

Thus 36 committees include may Josh.

(c) Does it seem likely Josh will be chosen for the committee if it is done randomly? Explain.

   Yes, even though it is done randomly there are some statistical chances that he will be selected in committee.

Thus,

Thus there are 3% chances that Josh is included in committee if it is done randomly.