Question

For any two sets $S$ and $T$, $S\Delta T$ is defined as the set of all elements that belong to either $S$ or $T$ but not both, that is $S\Delta T = \left( {S \cup T} \right) - \left( {S \cap T} \right)$. Let $A$, $B$ and $C$ be sets such that $A \cap B \cap C = \phi$, and the number of elements in each of $A\Delta B$, $B\Delta C$ and $C\Delta A$ equals to 100. Then the number of elements in $A \cup B \cup C$ equals

Answer 1

We will use the formula, $n\left( {A \cup B \cup C} \right) = n\left( A \right) + n\left( B \right) + n\left( C \right) - n\left( {A \cap B} \right) - n\left( {B \cap C} \right) - n\left( {C \cap A} \right) + n\left( {A \cap B \cap C} \right)$ to find the value of number of elements of$A \cup B \cup C$. We will then substitute the value of $A \cap B \cap C = \phi$. We will then use the Venn diagram to find the value of the unknowns.

Complete step-by-step answer:
We have to find the value of $A \cup B \cup C$
We know that $n\left( {A \cup B \cup C} \right) = n\left( A \right) + n\left( B \right) + n\left( C \right) - n\left( {A \cap B} \right) - n\left( {B \cap C} \right) - n\left( {C \cap A} \right) + n\left( {A \cap B \cap C} \right)$
Also, we are given that $A \cap B \cap C = \phi$ and $n\left( {A \cap B \cap C} \right) = 0$
$n\left( {A \cup B \cup C} \right) = n\left( A \right) + n\left( B \right) + n\left( C \right) - n\left( {A \cap B} \right) - n\left( {B \cap C} \right) - n\left( {C \cap A} \right)$ eqn. (1)
We are also given that
$n\left( {A\Delta B} \right) = 100 \\\ \Rightarrow n\left( {\left( {A \cup B} \right) - \left( {A \cap B} \right)} \right) = 100 \\\$
Let us now simplify the expression $\left( {A \cup B} \right) - \left( {A \cap B} \right)$ using Venn-Diagram.
If $A$ and $B$ are two sets, then the intersection includes only the common portion and the union includes all the elements of both the sets.

Hence, $n\left( {\left( {A \cup B} \right) - \left( {A \cap B} \right)} \right) = n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)$
Similarly, we can write this for any two sets.
Therefore, we can write equation (1) as
$n\left( {A \cup B \cup C} \right) = n\left( A \right) + n\left( B \right) + n\left( C \right) - n\left( {A \cap B} \right) - n\left( {B \cap C} \right) - n\left( {C \cap A} \right) \\\ \Rightarrow n\left( {A \cup B \cup C} \right) = \left( {n\left( A \right) + n\left( B \right) - n\left( {A \cap B} \right)} \right) + n\left( C \right) - n\left( {B \cap C} \right) - n\left( {C \cap A} \right) \\\$
Here, we do not have any condition on the number of intersections of two sets, therefore, $A \cup B \cup C$ cannot be determined.

Note: We use Venn diagrams to represent the sets. $A \cup B$ is the set of elements from both the sets $A$ and $B$, similarly, $A \cap B$ has all common elements of $A$ and $B$. The set $A - B$ has the elements of set $A$ which are not a part of $A \cap B$.