1. **List the elements of the sets:** **a)** $\{ x : x \in \mathbb{N}, 5 < x < 12 \}$ means natural numbers greater than 5 and less than 12. So, the elements are $6, 7, 8, 9, 10, 11$. **b)** $\{ x : x \in \mathbb{N}, x \text{ is even}, x < 15 \}$ means even natural numbers less than 15. The even numbers less than 15 are $2, 4, 6, 8, 10, 12, 14$. **c)** $\{ x : x \in \mathbb{N}, 10 < x < 35, \text{sum of digits} < 6 \}$ means natural numbers between 11 and 34 whose digits add up to less than 6. Check each number: - 11: $1+1=2 < 6$ include - 12: $1+2=3 < 6$ include - 13: $1+3=4 < 6$ include - 14: $1+4=5 < 6$ include - 15: $1+5=6 \not< 6$ exclude - 16: $1+6=7$ exclude - 17: $1+7=8$ exclude - 18: $1+8=9$ exclude - 19: $1+9=10$ exclude - 20: $2+0=2 < 6$ include - 21: $2+1=3 < 6$ include - 22: $2+2=4 < 6$ include - 23: $2+3=5 < 6$ include - 24: $2+4=6$ exclude - 25: $2+5=7$ exclude - 26: $2+6=8$ exclude - 27: $2+7=9$ exclude - 28: $2+8=10$ exclude - 29: $2+9=11$ exclude - 30: $3+0=3 < 6$ include - 31: $3+1=4 < 6$ include - 32: $3+2=5 < 6$ include - 33: $3+3=6$ exclude - 34: $3+4=7$ exclude So the elements are $11, 12, 13, 14, 20, 21, 22, 23, 30, 31, 32$. 2. **Given universal set $U = \{ x : x \text{ is an integer}, 1 \leq x \leq 10 \}$** **P = \{ x : x \text{ is odd} \}$** Odd numbers between 1 and 10 are $1, 3, 5, 7, 9$. **Q = \{ x : x \text{ is a factor of } 12 \}$** Factors of 12 are $1, 2, 3, 4, 6, 12$ but since $x \in U$, only $1, 2, 3, 4, 6$. **a)** Elements of $P$ are $\{1, 3, 5, 7, 9\}$. Elements of $Q$ are $\{1, 2, 3, 4, 6\}$. **b)** $n(P) = 5$ (number of elements in $P$). $n(Q) = 5$ (number of elements in $Q$). 3. **List all elements of the following sets:** **a)** $A = \{ x : x \text{ is a letter in the word MATHEMATICS} \}$ The letters are M, A, T, H, E, M, A, T, I, C, S. Unique letters: $\{M, A, T, H, E, I, C, S\}$. **b)** $B = \{ x : x \text{ is a month of a year not having 31 days} \}$ Months without 31 days: February (28 or 29), April (30), June (30), September (30), November (30). So $B = \{\text{February}, \text{April}, \text{June}, \text{September}, \text{November}\}$. **c)** $C = \{ x : x \text{ is a letter before e in the English alphabet} \}$ Letters before e are a, b, c, d. **d)** $D = \{ x : x \text{ is a vowel in the word "EQUATION"} \}$ Vowels in EQUATION are E, U, A, I, O. **Final answers:** 1a) $\{6, 7, 8, 9, 10, 11\}$ 1b) $\{2, 4, 6, 8, 10, 12, 14\}$ 1c) $\{11, 12, 13, 14, 20, 21, 22, 23, 30, 31, 32\}$ 2a) $P = \{1, 3, 5, 7, 9\}$, $Q = \{1, 2, 3, 4, 6\}$ 2b) $n(P) = 5$, $n(Q) = 5$ 3a) $A = \{M, A, T, H, E, I, C, S\}$ 3b) $B = \{\text{February}, \text{April}, \text{June}, \text{September}, \text{November}\}$ 3c) $C = \{a, b, c, d\}$ 3d) $D = \{E, U, A, I, O\}$