# Lesson 16 Solutions

Exercise 1) **How many elements are in the union of the sets and ? How many are in their intersection?**

Solution: The union of these two sets and is the set whose elements are either in or in . Thus, 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 , so the intersection has only 2 elements.

Exercise 2) **What is the union of the sets and ? What is their intersection?**

Solution: If we let and , 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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On to Lesson 17

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