We are given that A and B are sets such that $A \cap X = B \cap X = \phi$ and $A \cup X = B \cup X$. We need to show that A = B. We will use the hint provided: $A = A \cap (A \cup X)$ and $B = B \cap (B \cup X)$.

First, let us prove that A is a subset of B. We start with the expression for A:

$$A = A \cap (A \cup X)$$

Since $A \cup X = B \cup X$ (given), we can substitute this into the equation:

$$A = A \cap (B \cup X)$$

Using the distributive law, we distribute the intersection over the union:

$$A = (A \cap B) \cup (A \cap X)$$

We are given that $A \cap X = \phi$. Substituting this value:

$$A = (A \cap B) \cup \phi$$ $$A = A \cap B$$

Since $A = A \cap B$, it implies that A is a subset of B ($A \subset B$).

Next, let us prove that B is a subset of A. We start with the expression for B:

$$B = B \cap (B \cup X)$$

Since $B \cup X = A \cup X$ (given), we substitute this into the equation:

$$B = B \cap (A \cup X)$$

Using the distributive law:

$$B = (B \cap A) \cup (B \cap X)$$

We are given that $B \cap X = \phi$. Substituting this value:

$$B = (B \cap A) \cup \phi$$ $$B = B \cap A$$

Since $B = B \cap A$, it implies that B is a subset of A ($B \subset A$).

Since we have shown that $A \subset B$ and $B \subset A$, we can conclude that the sets are equal. Therefore, A = B is proven.