If B is the set whose elements are obtained by adding 1 to each of the even numbers , then the set builder notation of B is

Options

A. B= [x: x is even ]
B. B= [X:x is odd and x?1]
C. B={x:x is odd and x `in ` z }
D. B= { x:x is an interger }

Explanation

To find the set builder notation for the set \( B \), whose elements are obtained by adding 1 to each of the even numbers, we can follow these steps:

### Step-by-Step Solution:

1. **Identify Even Numbers**:
Even numbers can be expressed in the form \( 2n \), where \( n \) is an integer. This is because even numbers are multiples of 2.

2. **Add 1 to Even Numbers**:
If we add 1 to each even number, we get \( 2n + 1 \). This expression represents all odd numbers since adding 1 to an even number results in an odd number.

3. **Define the Set B**:
The set \( B \) consists of all elements of the form \( 2n + 1 \). Therefore, we can express this in set builder notation.

4. **Set Builder Notation**:
The set builder notation for \( B \) can be written as:
\[ B = \{ x \mid x = 2n + 1, n \in \mathbb{Z} \} \]
Alternatively, since \( x \) represents odd numbers, we can also write:
\[ B = \{ x \mid x \text{ is an odd integer} \} \]

5. **Conclusion**:
The final set builder notation for set \( B \) is:
\[ B = \{ x \mid x \text{ is an odd integer}, x \in \mathbb{Z} \} \]