Lesson 16 Solutions

Exercise 1)  How many elements are in the union of the sets \{1, 2, 3, 4, 7\} and {\{3, 4, 5\}}? How many are in their intersection?

Solution: The union of these two sets \{1, 2, 3, 4, 7\} and \{3, 4, 5\} is the set whose elements are either in \{1,2, 3, 4, 7\} or in \{3, 4, 5\}.  Thus, \{1, 2, 3, 4, 7\} \cup \{3, 4, 5\} = \{1, 2, 3, 4, 5, 7\} and so there are 6 elements in the union.  The intersection of these two sets is the set whose elements are in both of these sets, so \{1, 2, 3, 4, 7\} \cap \{3, 4, 5\} = \{3, 4\}, so the intersection has only 2 elements.

Exercise 2) What is the union of the sets {\{p\in \mathbb{N} | p \mathrm{\ is\ even}\}} and {\{p\in \mathbb{N}| p \mathrm{\ is\ a\ multiple\ of\ 4}\}}? What is their intersection?

Solution:  If we let A=\{p\in \mathbb{N} | p \mathrm{\ is\ even}\} and B=\{p\in \mathbb{N}| p \mathrm{\ is\ a\ multiple\ of\ 4}\}, then we can say that B is a subset of A, because everything that is in B is also in A.  Thus, the union of these two sets is simply A because B doesn’t “bring anything new to the table”, as all of its elements are already in A.  Similarly, the intersection of A and B is simply B, because anything in B is in both A and B, which is the definition of the intersection of two sets.  This is a general phenomena: if B is a subset of A, then A union B is A, and A intersect B is B.

Exercise 3)  What is the union of any set A with the empty set? What is their intersection?

Solution:  For this we simply steal the results from exercise 2.  Namely, since the empty set is a subset of every set, then the union of any set A with the empty set is simply A again (because the empty set most surely doesn’t “bring anything new to the table”).  Similarly, the intersection of any set A with the empty set is the empty set, because there is nothing that is in both A and the empty set, simply because there is nothing in the empty set to begin with!

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