If A, B and C are three sets such that A ∩ B = A ∩ C and A ∪ B = A ∪ C. then

The correct option is B = C.

To understand why this is correct, let us prove that the sets $B$ and $C$ are equal using the properties of set operations.

We are given two conditions:
1) $A\cap B=A\cap C$
2) $A\cup B=A\cup C$

Let us start with the set $B$. By the absorption law of set theory, we can write $B$ as:
$B=B\cap (A\cup B)$

Using the second given condition ($A\cup B=A\cup C$), we can substitute $A\cup C$ into our equation:
$B=B\cap (A\cup C)$

Next, we apply the distributive law of intersection over union:
$B=(B\cap A)\cup (B\cap C)$

Since intersection is commutative, $B\cap A=A\cap B$:
$B=(A\cap B)\cup (B\cap C)$

Now, we use the first given condition ($A\cap B=A\cap C$) to substitute $A\cap C$ for $A\cap B$:
$B=(A\cap C)\cup (B\cap C)$

Using the distributive law in reverse (factoring out the intersection with $C$), we get:
$B=(A\cup B)\cap C$

Again, using the second given condition ($A\cup B=A\cup C$), we substitute $A\cup C$ in place of $A\cup B$:
$B=(A\cup C)\cap C$

Finally, by applying the absorption law again for set $C$, we know that $(A\cup C)\cap C=C$. Therefore, we obtain:
$B=C$

Hence, the relationship $B=C$ is mathematically proven to be correct.