Suppose A , B and C are three sets such that `A cup C = B cup C` and `A cap C = B cap C`, then

Options

A. `A = B`
B. `A = B = phi`
C. A = B = C
D. `A cup B sube C`

Answer

To solve the problem, we need to analyze the given conditions:

1. **Given**: \( A \cup C = B \cup C \)
2. **Given**: \( A \cap C = B \cap C \)

We need to show that these conditions imply \( A = B \).

### Step 1: Start with the first condition
From the first condition \( A \cup C = B \cup C \), we can express set \( B \) in terms of sets \( A \) and \( C \).

### Step 2: Rewrite set \( B \)
We can express \( B \) as follows:
\[ B = (B \cup C) \cap B \]
Using the first condition, we substitute:
\[ B = (A \cup C) \cap B \]

### Step 3: Apply distributive property
Using the distributive property of sets, we can write:
\[ B = (A \cup C) \cap B = (A \cap B) \cup (C \cap B) \]

### Step 4: Use the second condition
Now, we apply the second condition \( A \cap C = B \cap C \):
From this, we can say:
\[ C \cap B = A \cap C \]

### Step 5: Substitute back into the expression for \( B \)
Now substituting \( C \cap B \) in the expression for \( B \):
\[ B = (A \cap B) \cup (A \cap C) \]

### Step 6: Simplify the expression
Since \( A \cap C = B \cap C \), we can rewrite:
\[ B = (A \cap B) \cup (B \cap C) \]

### Step 7: Analyze the expression
This means that every element of \( B \) must also be in \( A \) because \( B \) can be expressed in terms of \( A \) and \( C \).

### Step 8: Conclude that \( A = B \)
Thus, we conclude that:
\[ B \subseteq A \]
And since we started with \( A \cup C = B \cup C \) and \( A \cap C = B \cap C \), we can also show that:
\[ A \subseteq B \]
Therefore, we can conclude:
\[ A = B \]

### Final Conclusion
Thus, the final result is:
\[ A = B \]