If U= {x:x is a letter in English alphabet}
A={x:x is a vowel in English alphabet }.
Find `A^(c)and(A^(c))^(c)` .

Answer

To solve the problem, we need to find the complement of set A and then the complement of that result. Let's go through the steps systematically.

### Step 1: Define the Universal Set (U)
The universal set U contains all the letters in the English alphabet.
Thus,
\[ U = \{ a, b, c, d, e, f, g, h, i, j, k, l, m, n, o, p, q, r, s, t, u, v, w, x, y, z \} \]

### Step 2: Define Set A
Set A contains all the vowels in the English alphabet.
Thus,
\[ A = \{ a, e, i, o, u \} \]

### Step 3: Find the Complement of Set A (A^c)
The complement of set A, denoted as \( A^c \), includes all elements in the universal set U that are not in set A.
To find \( A^c \):
\[ A^c = U - A \]
This means we will take all the letters in U and remove the vowels (elements in A).

Calculating \( A^c \):
\[ A^c = \{ b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z \} \]

### Step 4: Find the Complement of A^c ((A^c)^c)
Now, we need to find the complement of \( A^c \), denoted as \( (A^c)^c \). This will include all elements in U that are not in \( A^c \).

Calculating \( (A^c)^c \):
Since \( A^c \) contains all consonants, the complement will bring back the vowels:
\[ (A^c)^c = A = \{ a, e, i, o, u \} \]

### Final Results
- \( A^c = \{ b, c, d, f, g, h, j, k, l, m, n, p, q, r, s, t, v, w, x, y, z \} \)
- \( (A^c)^c = \{ a, e, i, o, u \} \)