1.
Give the definition for these terms. Be sure to set up any notation that is required.
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set
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equal sets
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empty set
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subset
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proper subset
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cardinality of a set
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power set of a set
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Cartesian product of sets
Exercises 2.1.6 Exercises
2.
Write each set in roster notation.
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\(\displaystyle \set{n \in \ZZ \given n\text{ is odd and }n\leq 10}\)
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\(\displaystyle \set{n \in \ZZ \given \abs{n}\leq 3}\)
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the set of positive, even integers less than \(6\)
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\(\displaystyle \set{x \in \RR \given x^2 - x - 1 = 0}\)
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\(\displaystyle \set{x \in \RR \given x\text{ is the square of an integer and }x \lt 100}\)
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\(\displaystyle \set{x \in \RR \given x^2 = -1}\)
3.
Use set builder notation to give a description of each of these sets.
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\(\displaystyle \set{2, 3, 4, 5, 6, 7, 8}\)
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\(\displaystyle \set{0, 3, 6, 9, 12}\)
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\(\displaystyle \set{-2, -1, 0, 1, 2}\)
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\(\displaystyle \set{0, 4, 9, 16, 25, 36}\)
4.
For each of these pairs of sets, determine whether the first is a subset of the second, the second is a subset of the first, or neither is a subset of the other.
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the set of dogs; the set of mammals
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the set of people who speak English; the set of people who
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the set of math majors; the set of computer science majors
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the set of animals; the set of alligators
5.
For each of these sets, determine whether \(3\) is an element of that set.
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\(\displaystyle \set{1, 3, 5, 7, 9}\)
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\(\displaystyle \set{\set{3, 5}, \set{2, 4}}\)
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\(\displaystyle \set{(0, 3), (3, 0), (3, 3)}\)
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\(\displaystyle \set{\set{3}, \set{5}}\)
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the set of positive, odd integers less than \(20\)
6.
Find \(A \times B\text{,}\) where \(A = \set{a, b}\) and \(B = \set{0,4,8}\text{.}\)
What is the cardinality of each of these sets?
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\(\displaystyle \set{1, 3, 3, 5, 5, 6, 10, 10}\)
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\(\displaystyle \set{\set{1, 3, 5, 6}, \set{10, 12, 15}}\)
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\(\displaystyle \emptyset\)
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\(\displaystyle \set{0, \emptyset}\)
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\(\displaystyle \set{\emptyset, \set{}}\)
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\(\displaystyle \cP(\set{0, 1})\)
7.
Suppose \(A\text{,}\) \(B\text{,}\) and \(C\) are sets such that \(A \subseteq B\) and \(B \subseteq C\text{.}\) Prove that \(A \subseteq C\text{.}\)
8.
Let \(A\text{,}\) \(B\text{,}\) \(C\text{,}\) and \(D\) be sets. Show that if \(A \subseteq C\) and \(B \subseteq D\text{,}\) then \(A \times B \subseteq C \times D\text{.}\)
9.
Let \(A = \set{a, b, c}\text{,}\) and let \(B = \set{x, y}\text{.}\) Find the following sets.
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\(\displaystyle A\times B\)
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\(\displaystyle B \times A\)
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\(\displaystyle A^2\)
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\(B^3\text{.}\)
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\(\displaystyle \cP(A)\)
10.
Determine whether these statements are true or false.
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\(\displaystyle x \in \set{x}\)
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\(\displaystyle \set{x} \in \set{x}\)
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\(\displaystyle x \subseteq \set{x}\)
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\(\displaystyle \set{x} \subseteq \set{x}\)
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\(\displaystyle \emptyset \subseteq \set{x}\)
