Try It Now

Work these exercises to see how well you understand this material.

List all partitions of the set A = {a, b, c}.
A student, on an exam paper, defined the term partition the following way: “Let A be a set. A partition of A is any set of nonempty subsets A₁, A₂, A₃, . . . of A such that each element of A is in one of the subsets.” Is this definition correct? Why?
Show that {{2n| n ∈ ℤ},{2n+ 1 | n ∈ ℤ}} is a partition of ℤ. Describe this partition using only words.
A survey of 90 people, 47 of them played tennis and 42 of them swam. If 17 of them participated in both activities, how many of them participated in neither?
Regarding the Theorem 2.3.9,
1. (a) Use the two set inclusion-exclusion law to derive the three set inclusion- exclusion law. Note: A knowledge of basic set laws is needed for this exercise.
2. (b) State and derive the inclusion-exclusion law for four sets.
The definition of ℚ = {a/b| a, b ∈ ℤ, b ≠ 0} given in Chapter 1 is awkward. If we use the definition to list elements in ℚ, we will have duplications such as $\frac{1}{2}$ , $\frac{-2}{-4}$ , and $\frac{300}{600}$ . Try to write a more precise definition of the rational numbers so that there is no duplication of elements.