If `A DeltaB = {1,2,3,4,5,6,7,8,9,11,13}` and `B = {4,5,6,8,11,13}` then find A.

Options

A. `{1,2,3,7,9}`
B. `{1,2,3,4,5,6,7,9}`
C. `{1,2,3,4,6,7,9}`
D. `{1,2,3,4,8,11}`

Explanation

To find the set A given that \( A \Delta B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13\} \) and \( B = \{4, 5, 6, 8, 11, 13\} \), we can follow these steps:

### Step 1: Understand the definition of symmetric difference
The symmetric difference \( A \Delta B \) is defined as:
\[ A \Delta B = (A \cup B) - (A \cap B) \]
This means it includes all elements that are in either A or B but not in both.

### Step 2: Identify elements in \( A \Delta B \)
We are given:
\[ A \Delta B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13\} \]
This set contains elements that are either in A or in B, but not in both.

### Step 3: Identify elements in set B
We have:
\[ B = \{4, 5, 6, 8, 11, 13\} \]
These elements are part of the union but will not be part of the intersection if they are also in A.

### Step 4: Determine elements in A
To find A, we need to identify which elements from \( A \Delta B \) are not in B. We will subtract the elements of B from \( A \Delta B \):
- Elements in \( A \Delta B \) that are not in B:
- From \( \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13\} \), we remove \( \{4, 5, 6, 8, 11, 13\} \).

Thus, the remaining elements are:
\[ \{1, 2, 3, 7, 9\} \]

### Step 5: Combine with elements of B to find A
Since the elements \( \{4, 5, 6, 8, 11, 13\} \) are in B, we also need to consider that A must include these elements as well. Therefore, we can conclude:
\[ A = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13\} \]

### Final Answer
Thus, the set A is:
\[ A = \{1, 2, 3, 7, 9\} \cup B = \{1, 2, 3, 4, 5, 6, 7, 8, 9, 11, 13\} \]