1. **Problem:** Is there a real number whose square root is -1? a. Is there a real number $x$ such that $\sqrt{x} = -1$? b. Does there exist $x$ such that $\sqrt{x} = -1$? **Explanation:** The square root function $\sqrt{x}$ for real numbers $x \geq 0$ always returns a non-negative number. Therefore, $\sqrt{x} = -1$ has no solution in real numbers. **Answer:** No real number $x$ satisfies $\sqrt{x} = -1$. 2. **Problem:** Given any real number, there is a real number that is lesser. a. Given any real number $r$, there is $s$ such that $s < r$. b. For any real number $r$, there exists $s$ such that $s < r$. **Explanation:** The real numbers are dense and unbounded below, so for any real number $r$, you can find another real number $s$ less than $r$ (for example, $s = r - 1$). 3. **Problem:** For all real numbers $x$, if $x$ is an integer then $x$ is a rational number. a. If a real number is an integer, then it is rational. b. For all integers $x$, $x$ is rational. c. If $x = n$ where $n \in \mathbb{Z}$, then $x$ is rational. d. All integers $x$ are rational numbers. **Explanation:** Every integer can be expressed as a ratio of itself over 1, so integers are rational. 4. **Problem:** All real numbers have squares that are not equal to -1. a. Every real number has a square $\geq 0$. b. For all real numbers $r$, $r^2 \neq -1$. c. For all real numbers $r$, there is a real number $s = r^2$ such that $s \neq -1$. **Explanation:** Squares of real numbers are always non-negative, so they cannot be -1. 5. **Problem:** There is a positive integer whose square is equal to itself. a. Some positive integer $n$ has the property that $n^2 = n$. b. There is a real number $r$ such that $r^2 = r$. c. There is a real number $r$ with the property that for every real number $s$, $r^2 = r$. **Explanation:** The positive integers 1 satisfies $1^2 = 1$. 6. **Problem:** Let $A$ be the set containing all prime numbers less than 30. a. $A = \{2, 3, 5, 7, 11, 13, 17, 19, 23, 29\}$. b. Is $\{2, 2\} = \{2, (2)\}$? Yes, because sets do not count duplicates or order. c. The set $\{a, a, a, a, a\}$ has 1 element because duplicates are not counted. 7. **Problem:** Describe the sets. a. $\{x \in \mathbb{N} \mid x \leq 10 \text{ and } x \text{ divisible by } 3\} = \{3, 6, 9\}$. b. $\{x \in \mathbb{Z} \mid x \text{ is prime and divisible by } 2\} = \{2\}$. c. $\{x \in \mathbb{Z} \mid x^2 = 4\} = \{-2, 2\}$. 8. **Problem:** Given $B = \{2,4,6,8,10\}$, $C = \{4,8,10\}$, $D = \{x \mid x \text{ is even}\}$. a. Is $D \subset B$? No, because $D$ contains even numbers not in $B$ (e.g., 12). b. Is $C \subset D$? Yes, all elements of $C$ are even. c. Is $C \subset B$? Yes, all elements of $C$ are in $B$. d. Is $B$ a proper subset of $D$? Yes, because $B \subset D$ and $B \neq D$. 9. **Problem:** Compare ordered pairs. a. Is $((-1)^2, 1^2) = (1^2, (-1)^2)$? Yes, both equal $(1,1)$. b. Is $(\sqrt{16}, 1/4) = (4, 3/12)$? Yes, $\sqrt{16} = 4$ and $1/4 = 3/12 = 0.25$. c. Is $(-2^2, 0) = (-\sqrt{16}, 0)$? No, $-2^2 = -(2^2) = -4$ but $-\sqrt{16} = -4$, so yes they are equal. 10. **Problem:** Let $A = \{1,2,3,4\}$ and $B = \{0,1\}$. a. $A \times B = \{(1,0),(1,1),(2,0),(2,1),(3,0),(3,1),(4,0),(4,1)\}$ with 8 elements. b. $B \times A = \{(0,1),(0,2),(0,3),(0,4),(1,1),(1,2),(1,3),(1,4)\}$ with 8 elements. c. $A \times A$ has $4 \times 4 = 16$ elements. d. $B \times B$ has $2 \times 2 = 4$ elements. 11. **Problem:** Relation $R$ from $C = \{0,1,2\}$ to $D = \{2,4,6,8\}$ defined by $(x,y) \in R$ if $(y+2)/x$ is integer. a. Check if $1 R 2$: $(2+2)/1=4$ integer, yes. Check if $2 R 8$: $(8+2)/2=5$ integer, yes. Check if $(1,8) \in R$: same as $1 R 8$, $(8+2)/1=10$ integer, yes. Check if $(2,6) \in R$: $(6+2)/2=4$ integer, yes. b. $R = \{(1,2),(1,4),(1,6),(1,8),(2,2),(2,4),(2,6),(2,8)\}$ but note division by zero is undefined for $x=0$. c. Domain: $\{1,2\}$ (since $x=0$ causes division by zero), Codomain: $\{2,4,6,8\}$. 12. **Problem:** Relation $A$ from $\mathbb{R}$ to $\mathbb{R}$ defined by $(x,y) \in A$ if $x = y$. a. Is $57.4 A 53$? No, $57.4 \neq 53$. Is $-17 A -14$? No. Is $(14,14) \in A$? Yes. Is $(-35,1) \in A$? No. b. The graph of $A$ is the line $y = x$ in the Cartesian plane. 13. **Problem:** Relations and functions from $\{a,b,c\}$ to $\{u,v\}$. a. Number of relations: $2^{3 \times 2} = 2^6 = 64$. b. Number of functions: each of 3 elements maps to 2 choices, so $2^3 = 8$. c. Fraction of relations that are functions: $\frac{8}{64} = \frac{1}{8}$. **Final answers:** - No real $x$ satisfies $\sqrt{x} = -1$. - For any real $r$, there exists $s$ such that $s < r$. - Integers are rational. - Squares of real numbers are never -1. - Positive integer 1 satisfies $n^2 = n$. - $A = \{2,3,5,7,11,13,17,19,23,29\}$. - Sets ignore duplicates. - Descriptions of sets as above. - Subset relations as above. - Ordered pairs equality as above. - Cartesian products and their sizes as above. - Relation $R$ and domain/codomain as above. - Relation $A$ is equality, graph is $y=x$. - Number of relations and functions and fraction as above.