To find [tex]\( A^c \)[/tex], we will first clearly define our universal set [tex]\( U \)[/tex] and our subset [tex]\( A \)[/tex].
1. The universal set [tex]\( U \)[/tex] includes all the letters in the English alphabet:
[tex]\[ U = \{a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z\} \][/tex]
2. The subset [tex]\( A \)[/tex] contains all the consonants in the English alphabet. Consonants are all letters that are not vowels:
[tex]\[ A = \{b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z\} \][/tex]
3. To find [tex]\( A^c \)[/tex] (the complement of [tex]\( A \)[/tex]), we identify all the elements in [tex]\( U \)[/tex] that are not in [tex]\( A \)[/tex]. Essentially, [tex]\( A^c \)[/tex] consists of the vowels:
[tex]\[ A^c = U \setminus A \][/tex]
The vowels in the English alphabet are:
[tex]\[ A^c = \{a, e, i, o, u\} \][/tex]
So the complement of [tex]\( A \)[/tex], in proper set notation, is:
[tex]\[ \{a, e, i, o, u\} \][/tex]
