Let U= {1,2,3,4,5,6,},A {2,3} and
B={3,4,5} . Find A ,B,A `capB',AuuBand` hence show that `(AuuB)'=A'capB`.

Answer

To solve the problem step by step, we will follow the instructions given in the question.

### Step 1: Identify the Universal Set and the Given Sets
- Let \( U = \{1, 2, 3, 4, 5, 6\} \)
- Let \( A = \{2, 3\} \)
- Let \( B = \{3, 4, 5\} \)

### Step 2: Find the Complement of Set A
The complement of set \( A \) (denoted as \( A' \)) is found by subtracting the elements of \( A \) from the universal set \( U \).
- \( A' = U - A = \{1, 2, 3, 4, 5, 6\} - \{2, 3\} = \{1, 4, 5, 6\} \)

### Step 3: Find the Complement of Set B
Similarly, the complement of set \( B \) (denoted as \( B' \)) is found by subtracting the elements of \( B \) from the universal set \( U \).
- \( B' = U - B = \{1, 2, 3, 4, 5, 6\} - \{3, 4, 5\} = \{1, 2, 6\} \)

### Step 4: Find the Intersection of Complements \( A' \cap B' \)
Now, we will find the intersection of \( A' \) and \( B' \).
- \( A' \cap B' = \{1, 4, 5, 6\} \cap \{1, 2, 6\} = \{1, 6\} \)

### Step 5: Find the Union of Sets A and B
Next, we will find the union of sets \( A \) and \( B \) (denoted as \( A \cup B \)).
- \( A \cup B = \{2, 3\} \cup \{3, 4, 5\} = \{2, 3, 4, 5\} \)

### Step 6: Find the Complement of the Union \( (A \cup B)' \)
Now, we will find the complement of the union \( A \cup B \).
- \( (A \cup B)' = U - (A \cup B) = \{1, 2, 3, 4, 5, 6\} - \{2, 3, 4, 5\} = \{1, 6\} \)

### Step 7: Show that \( (A \cup B)' = A' \cap B' \)
From the previous steps, we found:
- \( (A \cup B)' = \{1, 6\} \)
- \( A' \cap B' = \{1, 6\} \)

Since both results are equal, we can conclude that:
- \( (A \cup B)' = A' \cap B' \)

### Final Conclusion
We have shown that the complement of the union of sets \( A \) and \( B \) is equal to the intersection of their complements.

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