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Let [tex]$U$[/tex] be the universal set consisting of all the letters in the English alphabet. Let [tex]$A$[/tex] be the subset of [tex][tex]$U$[/tex][/tex] consisting of all the consonants. Find [tex]$A^c$[/tex]. Write your answer in proper set notation, for example [tex]\{a, b, c, d, e\}[/tex].

Provide your answer below:

Sagot :

GinnyAnswer
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]
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