If `U={11,12,13,14,....,20},A={11,12,13,14} and B={14,16,18,20}`, then find each of the following :
(i) `A, cap B'`
(ii) `A' cup B'`
(iii) `(A cup B)'`
(iv) `(A cap B)'`
(v) `A ' - B'`
(vi) `B - A'`.

Answer

To solve the given problem step by step, we will first define the sets and then find the required operations.

### Given Sets:
- Universal Set \( U = \{11, 12, 13, 14, 15, 16, 17, 18, 19, 20\} \)
- Set \( A = \{11, 12, 13, 14\} \)
- Set \( B = \{14, 16, 18, 20\} \)

### Step 1: Find the complements of sets A and B
- The complement of \( A \) (denoted as \( A' \)) is the set of elements in \( U \) that are not in \( A \):
\[ A' = U - A = \{15, 16, 17, 18, 19, 20\} \]

- The complement of \( B \) (denoted as \( B' \)) is the set of elements in \( U \) that are not in \( B \):
\[ B' = U - B = \{11, 12, 13, 15, 17, 19\} \]

### Step 2: Solve each part of the question

#### (i) \( A \cap B' \)
To find \( A \cap B' \), we need the intersection of sets \( A \) and \( B' \):
\[ A \cap B' = \{11, 12, 13, 14\} \cap \{11, 12, 13, 15, 17, 19\} = \{11, 12, 13\} \]

#### (ii) \( A' \cup B' \)
To find \( A' \cup B' \), we take the union of \( A' \) and \( B' \):
\[ A' \cup B' = \{15, 16, 17, 18, 19, 20\} \cup \{11, 12, 13, 15, 17, 19\} = \{11, 12, 13, 15, 16, 17, 18, 19, 20\} \]

#### (iii) \( (A \cup B)' \)
First, we find \( A \cup B \):
\[ A \cup B = \{11, 12, 13, 14\} \cup \{14, 16, 18, 20\} = \{11, 12, 13, 14, 16, 18, 20\} \]
Now, we find the complement:
\[ (A \cup B)' = U - (A \cup B) = \{15, 17, 19\} \]

#### (iv) \( (A \cap B)' \)
First, we find \( A \cap B \):
\[ A \cap B = \{11, 12, 13, 14\} \cap \{14, 16, 18, 20\} = \{14\} \]
Now, we find the complement:
\[ (A \cap B)' = U - (A \cap B) = \{11, 12, 13, 15, 16, 17, 18, 19, 20\} \]

#### (v) \( A' - B' \)
To find \( A' - B' \), we take the difference:
\[ A' - B' = \{15, 16, 17, 18, 19, 20\} - \{11, 12, 13, 15, 17, 19\} = \{16, 18, 20\} \]

#### (vi) \( B - A' \)
To find \( B - A' \), we take the difference:
\[ B - A' = \{14, 16, 18, 20\} - \{15, 16, 17, 18, 19, 20\} = \{14\} \]

### Summary of Results:
1. \( A \cap B' = \{11, 12, 13\} \)
2. \( A' \cup B' = \{11, 12, 13, 15, 16, 17, 18, 19, 20\} \)
3. \( (A \cup B)' = \{15, 17, 19\} \)
4. \( (A \cap B)' = \{11, 12, 13, 15, 16, 17, 18, 19, 20\} \)
5. \( A' - B' = \{16, 18, 20\} \)
6. \( B - A' = \{14\} \)