Sure, let's write these sets in set builder notation:
### (a) Students in your class.
[tex]\[ \text{students} = \{ x \mid x \text{ is a student in my class} \} \][/tex]
In this case, \{ Alice, Bob, Charlie, Daisy \}
### (b) Letters in the English alphabet.
[tex]\[ \text{alphabet} = \{ x \mid x \text{ is a letter in the English alphabet} \} \][/tex]
In this case, \{A, B, C, D, E, F, G, H, I, J, K, L, M, N, O, P, Q, R, S, T, U, V, W, X, Y, Z\}
### (c) [tex]\( A = \{2, 4, 6, 8\} \)[/tex]
[tex]\[ A = \{ x \in \mathbb{Z} \mid x = 2n \text{ where } 1 \leq n \leq 4 \} \][/tex]
### (d) [tex]\( B = \{3, 9, 27, 81\} \)[/tex]
[tex]\[ B = \{ x \in \mathbb{Z} \mid x = 3^n \text{ where } 1 \leq n \leq 4 \} \][/tex]
### (e) [tex]\( C = \{1, 4, 9, 16, 25\} \)[/tex]
[tex]\[ C = \{ x \in \mathbb{Z} \mid x = n^2 \text{ where } 1 \leq n \leq 5 \} \][/tex]
### (f) [tex]\( D = \{1, 3, 5, \ldots \} \)[/tex]
[tex]\[ D = \{ x \in \mathbb{Z} \mid x = 2n - 1 \text{ where } n \geq 1 \} \][/tex]
### (g) [tex]\( F = \{-10, \ldots , -3, -2, -1, 0, 1, 2, \ldots \} \)[/tex]
[tex]\[ F = \{ x \in \mathbb{Z} \mid -10 \leq x \leq 10 \} \][/tex]
### (h) [tex]\( G = \{ O \} \)[/tex]
[tex]\[ G = \{ x \mid x = O \} \][/tex]
### (i) [tex]\( P = \{ \} \)[/tex]
[tex]\[ P = \{ x \mid \text{False} \} \][/tex]
### (j) [tex]\( H = \{-5, 5\} \)[/tex]
[tex]\[ H = \{ x \in \mathbb{Z} \mid x = -5 \text{ or } x = 5 \} \][/tex]
Each of the above sets is presented in set builder notation, and they provide a formal way to describe all the elements that belong to those sets.
