`(Auuvarphi^(c))^(c)=` _________

Answer

To solve the expression \((A \cup \varphi^{c})^{c}\), we will follow these steps:

### Step 1: Understand the components
- Here, \(\varphi\) represents the empty set.
- The complement of the empty set, \(\varphi^{c}\), is the universal set \(U\), which contains all possible elements.

**Hint:** Remember that the complement of a set includes all elements that are not in that set.

### Step 2: Substitute the complement of the empty set
- We can rewrite the expression as:
\[ (A \cup U)^{c} \]

**Hint:** The union of any set with the universal set is the universal set itself.

### Step 3: Simplify the union
- Since \(A \cup U = U\), we have:
\[ (U)^{c} \]

**Hint:** Think about what happens when you take the union of a set with all possible elements.

### Step 4: Find the complement of the universal set
- The complement of the universal set \(U\) is the empty set \(\varphi\):
\[ (U)^{c} = \varphi \]

**Hint:** The complement of a set containing everything is a set containing nothing.

### Final Answer
Thus, the final result is:
\[ (A \cup \varphi^{c})^{c} = \varphi \]

### Summary of Steps
1. Identify the complement of the empty set as the universal set.
2. Substitute the complement into the expression.
3. Simplify the union with the universal set.
4. Determine the complement of the universal set, which is the empty set.