Write in words how to read each of the following out loud. Then write the shorthand notation for each set. a.

$$\{ x \in U | x \in A$$

and

$$x \in B \}$$

, b.

$$\{ x \in U | x \in A$$

or

$$x \in B \}$$

, c.

$$\{ x \in U | x \in A$$

and

$$x \notin B \}$$

, d.

$$\{ x \in U | x \notin A \}$$

a. Is 4={4}? b. How many elements are in the set {3, 4, 3, 5}? c. How many elements are in the set {1, {1}, {1, {1}}}?

Find the orthogonal complement of $S^\perp .$

$$S= span \left \{ \begin{bmatrix} 0\\1\\-1\\1 \end{bmatrix} \right \}$$

a. Is

$$2 \in \{ 2 \} ?$$

b. How many elements are in the set {2, 2, 2, 2}? c. How many elements are in the set {0, {0}}? d. Is

$$\{ 0 \} \in \{ \{ 0 \} , \{ 1 \} \} ?$$

, e. Is

$$0 \in \{ \{ 0 \} , \{ 1 \} \} ?$$

3. a. Is 4 = {4}? b. How many elements are in the set {3, 4, 3, 5}? c. How many elements are in the set {1, {1}, {1,{1}}}?

Write in words how to read each of the following out loud. Then write each set using the symbols for union, intersection, set difference, or set complement.

a. $\{x \in U \mid x \in A$ and $x \in B\}$

b. $\{x \in U \mid x \in A$ or $x \in B\}$

c. $\{x \in U \mid x \in A$ and $x \notin B\}$

d. $\{x \in U \mid x \notin A\}$

For each integer n, let

$$T _ { n } = \left\{ n , n ^ { 2 } \right\}$$

. How many elements are in each of

$$T _ { 2 } , T _ { - 3 } , T _ { 1 }$$

and

$$T_0$$

? Justify your answers.

In each fill in the blanks to rewrite the given statement. There is a real number whose product with every real number equals zero. a. Some ___ has the property that its ___. b. There is a real number a such that the product of a ___. c. There is a real number a with the property that for every real number b, ___.

In each fill in the blanks to rewrite the given statement. Every positive number has a positive square root. a. All positive numbers ____. b. For any positive number e, there is ___ for e. c. For all positive numbers e, there is a positive number r such that ___.

Every nonzero real number has a reciprocal. a. All nonzero real numbers ___. b. For all nonzero real numbers r, there is ___ for r. c. For all nonzero real numbers r, there is a real number s such that ___.

Write each of the following sets by listing its elements within braces. $(a)A=\{n \in \mathbf{Z}:-4<n \leq 4\} \\ (b)B=\left\{n \in \mathbf{Z}: n^{2}<5\right\} \\ (c)C=\left\{n \in \mathbf{N}: n^{3}<100\right\}\\ (d)D=\left\{x \in \mathbf{R}: x^{2}-x=0\right\}\\ (e)E=\left\{x \in \mathbf{R}: x^{2}+1=0\right\}$

In each, fill in the blanks to rewrite the given statement. There is a real number whose product with every number leaves the number unchanged. a. Some ___ has the property that its ___. b. There is a real number r such that the product of r ____. c. There is a real number r with the property that for every real number s, ____.

Let sets R, S, and T be defined as follows:

$$R = \{ x \in \mathbf { Z } | x$$

is divisiable by 2],

$$S = \{ y \in \mathbf { Z } | y$$

is divisible by 3},

$$T = \{ z \in \mathbf { Z } | z$$

is divisible by 6}. a. Is

$$R \subseteq T ?$$

Explain. b. Is

$$T \subseteq R ?$$

Explain. c. Is

$$T \subseteq S ?$$

Explain.

In each fill in the blanks to rewrite the given statement. For all objects J, if J is a square then J has four sides. a. All squares ___. b. Every square ___. c. If an object is a square, then it ___. d. If J ___, then J ___. e. For all squares J, ___.