Find a verbal description for \(O = \{ 2n-1 \, | \, n \in \mathbb {N}\}\)

\(O\) is the odd natural numbers.
Find a roster description for \(X = \{ z^2 \, | \, z \in \mathbb {Z}\}\)

\(X = \{ 0, 1, 4, 9, 16, \ldots \}\)
Let \(A = \left \{ -3, -1.02, -\dfrac {3}{5}, 0.57, 1.\overline {23}, \sqrt {3}, 5.2020020002 \ldots , \dfrac {20}{10}, 117 \right \}\)

  1. List the elements of \(A\) which are natural numbers.

    \(\dfrac {20}{10} = 2\) and \(117\)

  2. List the elements of \(A\) which are irrational numbers.

    \(\sqrt {3}\) and \(5.2020020002\)

  3. Find \(A \cap \mathbb {Z}\)

    \(\left \{ -3, \dfrac {20}{10}, 117\right \}\)

  4. Find \(A \cap \mathbb {Q}\)

    \(\left \{ -3, -1.02, -\dfrac {3}{5}, 0.57, 1.\overline {23},\dfrac {20}{10}, 117 \right \}\)
Fill in the chart below.
Set of Real Numbers Interval Notation Region on the Real Number Line

{x | − 1 ≤ x < 5}

[0,3)

[Picture]

{x | − 5 < x ≤ 0}

(− 3,3)

[Picture]

{x |x ≤ 3}

(− ∞, 9)

[Picture]

{x |x ≥ − 3}

In these problems, find the indicated intersection or union and simplify if possible. Express your answers in interval notation.

\((-1,5] \cap [0,8)\)

\([0,5]\)
\((-1,1) \cup [0,6]\)

\((-1,6]\)
\((-\infty ,4]\cap (0,\infty )\)

\((0,4]\)
\((-\infty ,0) \cap [1,5]\)

\(\emptyset \)
\((-\infty , 0) \cup [1,5]\)

\((-\infty ,0) \cup [1,5]\)
\((-\infty , 5] \cap [5,8)\)

\(\left \{ 5\right \}\)
In Exercises writeintervalfirst - writeintervallast, write the set using interval notation.
\(\{x\,|\, x \neq 5 \}\)

\((-\infty , 5) \cup (5, \infty )\)
\(\{x\,|\, x \neq -1 \}\)

\((-\infty , -1) \cup (-1, \infty )\)
\(\{x\,|\, x \neq -3,\, 4 \}\)

\((-\infty , -3) \cup (-3, 4)\cup (4, \infty )\)
\(\{x\,|\, x \neq 0, \, 2 \}\)

\((-\infty , 0) \cup (0, 2)\cup (2, \infty )\)
\(\{x\,|\, x \neq 2, \, -2 \}\)

\((-\infty , -2) \cup (-2, 2)\cup (2, \infty )\)
\(\{x\,|\, x \neq 0,\, \pm 4 \}\)

\((-\infty , -4) \cup (-4, 0) \cup (0, 4) \cup (4, \infty )\)
\(\{x\,|\, x \leq -1 \, \text {or} \, x \geq 1 \}\)

\((-\infty , -1] \cup [1, \infty )\)
\(\{x\,|\, x < 3 \, \text {and} \, x \geq 2 \}\)

\([2, 3)\)
\(\{x\,|\, x \leq -3 \, \text {or} \, x > 0 \}\)

\((-\infty , -3] \cup (0, \infty )\)
\(\{x\,|\, x \leq 2 \, \text {and} \, x > 3 \}\)

\(\emptyset \)
\(\{x\,|\, x > 2 \, \text {or} \, x = \pm 1 \}\)

\(\{-1\} \cup \{1\} \cup (2, \infty )\)
\(\{x\,|\, 3 < x < 13 \, \text {and} \, x \neq 4 \}\)

\((3,4) \cup (4, 13)\)
For Exercises shadevennfirst - shadevennlast, use the blank Venn Diagram below with \(A\), \(B\), and \(C\) in it as a guide to help you shade the following sets.

[Picture]

\(A \cup C\)
[Picture] [Picture] [Picture] [Picture] [Picture]
\(B \cap C\)
[Picture] [Picture] [Picture] [Picture] [Picture]
\((A \cup B) \cup C\)
[Picture] [Picture] [Picture] [Picture] [Picture]
\((A \cap B) \cap C\)
[Picture] [Picture] [Picture] [Picture] [Picture]
\(A \cap (B \cup C)\)
[Picture] [Picture] [Picture] [Picture] [Picture]
\((A \cap B) \cup (A \cap C)\)
[Picture] [Picture] [Picture] [Picture] [Picture]
Explain how your answers to problems intoverunion and shadevennlast show \(A \cap (B \cup C) = (A \cap B) \cup (A \cap C)\). Phrased differently, this shows ‘intersection distributes over union.’ Discuss with your classmates if ‘union’ distributes over ‘intersection.’ Use a Venn Diagram to support your answer.
Show that \(A \subseteq B\) if and only if \(A \cup B = B\).
Let \(A = \{1,3,5,7,9\}, B = \{2,4,6,8,10\}, C = \{1,6,9\}\) and \(D = \{2,7,10\}\). Draw one Venn Diagram that shows all four of these sets. What sort of difficulties do you encounter?