Q.$A$, $B$ and $C$ are subsets of Universal Set $U$. If $A = \{2, 4, 6, 8, 12, 20\}$, $B = \{3, 6, 9, 12, 15\}$, $C = \{5, 10, 15, 20\}$ and $U$ is the set of all whole numbers, draw a Venn diagram showing the relation of $U$, $A$, $B$ and $C$.

The key idea is Set Membership and Venn Diagram Representation, which visually depicts the relationships and common elements between sets within a universal set.

Here is the essential reasoning:

1. Identify the elements in each set: $A = \{2, 4, 6, 8, 12, 20\}$, $B = \{3, 6, 9, 12, 15\}$, $C = \{5, 10, 15, 20\}$.
2. Calculate the intersections:
   • $A \cap B = \{6, 12\}$
   • $B \cap C = \{15\}$
   • $A \cap C = \{20\}$
   • $A \cap B \cap C = \emptyset$ (no element is common to all three sets).
3. Determine elements unique to each non-overlapping region:
   • Elements in $A$ only: $A \setminus (B \cup C) = \{2, 4, 8\}$
   • Elements in $B$ only: $B \setminus (A \cup C) = \{3, 9\}$
   • Elements in $C$ only: $C \setminus (A \cup B) = \{5, 10\}$ …

A Venn diagram is a powerful visual tool used to represent the relationships between different sets. Each set is typically represented by a circle, and the universal set $U$ is represented by a rectangle enclosing these circles. The overlapping regions show the elements common to the intersecting sets. The intuition is to systematically place each element into the specific region that accurately describes its membership across the given sets.

Here's how we construct the Venn diagram:

1. Understand the Universal Set and Given Sets

   The universal set $U$ is the set of all whole numbers. This means any element we consider must be a whole number, and any whole number not found in $A, B,$ or $C$ will reside outside the circles but within the $U$ rectangle.

   The given sets are:

   $A = \{2, 4, 6, 8, 12, 20\}$

   $B = \{3, 6, 9, 12, 15\}$

   $C = \{5, 10, 15, 20\}$

2. Identify Elements in the Innermost Intersection ($A \cap B \cap C$)

   This region represents elements common to all three sets. We look for elements present in $A$, $B$, and $C$.

   • Elements in $A$: $\{2, 4, 6, 8, 12, 20\}$
   • Elements in $B$: $\{3, 6, 9, 12, 15\}$
   • Elements in $C$: $\{5, 10, 15, 20\}$ Comparing these, we find no element that appears in all three lists. Therefore, $A \cap B \cap C = \emptyset$. This means the central region where all three circles overlap will be empty.

3. Identify Elements in Two-Set Intersections (Excluding the Three-Set Intersection)

   Next, we find elements common to exactly two sets. We must exclude any elements already placed in $A \cap B \cap C$. Since $A \cap B \cap C$ is empty, we simply find the pairwise intersections.

   • $A \cap B$ (elements common to $A$ and $B$ only):

     Elements in $A$: $\{2, 4, \underline{6}, 8, \underline{12}, 20\}$

     Elements in $B$: $\{3, \underline{6}, 9, \underline{12}, 15\}$

     The common elements are $\{6, 12\}$. Since these are not in $C$, they belong to the region $A \cap B \setminus C$.

   • $B \cap C$ (elements common to $B$ and $C$ only):

     Elements in $B$: $\{3, 6, 9, 12, \underline{15}\}$

     Elements in $C$: $\{5, 10, \underline{15}, 20\}$

     The common element is $\{15\}$. Since $15$ is not in $A$, it belongs to the region $B \cap C \setminus A$.

   • $A \cap C$ (elements common to $A$ and $C$ only):

     Elements in $A$: $\{2, 4, 6, 8, 12, \underline{20}\}$

     Elements in $C$: $\{5, 10, 15, \underline{20}\}$

     The common element is $\{20\}$. Since $20$ is not in $B$, it belongs to the region $A \cap C \setminus B$.

4. Identify Elements Unique to Each Set

   Now we find elements that belong to only one set, meaning they are not in any of the intersection regions we've identified.

   • Elements in $A$ only ($A \setminus (B \cup C)$):

     Start with $A = \{2, 4, 6, 8, 12, 20\}$.

     Remove elements found in $A \cap B$ (which are $\{6, 12\}$) and $A \cap C$ (which is $\{20\}$).

     Remaining elements: $\{2, 4, 8\}$.

   • Elements in $B$ only ($B \setminus (A \cup C)$):

     Start with $B = \{3, 6, 9, 12, 15\}$.

     Remove elements found in $A \cap B$ (which are $\{6, 12\}$) and $B \cap C$ (which is $\{15\}$).

     Remaining elements: $\{3, 9\}$.

   • Elements in $C$ only ($C \setminus (A \cup B)$):

     Start with $C = \{5, 10, 15, 20\}$.

     Remove elements found in $B \cap C$ (which is $\{15\}$) and $A \cap C$ (which is $\{20\}$).

     Remaining elements: $\{5, 10\}$.

5. Identify Elements in $U$ but Not in $A \cup B \cup C$

   The universal set $U$ is the set of all whole numbers. The union of $A, B, C$ is a finite set:

   $A \cup B \cup C = \{2, 3, 4, 5, 6, 8, 9, 10, 12, 15, 20\}$. …