To illustrate the relationship among the sets $U, A, B,$ and $C$ using a Venn diagram, we need to analyze each set:

1. Universal Set $U$: Contains all distinct letters in "MATHEMATICS". $U = \{M, A, T, H, E, I, C, S\}$
2. Set $A$: Given as $A = \{M, A, T, H\}$.
3. Set $B$: Given as $B = \{A, I, M\}$.
4. Set $C$: Given as $C = \{T, E, A, M\}$.

Analysis of Relationships:

• Common Elements:
  • $A \cap B = \{M, A\}$
  • $A \cap C = \{M, A, T\}$
  • $B \cap C = \{M, A\}$
  • $A \cap B \cap C = \{M, A\}$
• The remaining elements of $U$ (not in $A, B,$ or $C$) are $\{C, S\}$.

Venn Diagram Representation:

The Venn diagram will consist of:

• A rectangle representing $U$.
• Three overlapping circles for $A, B,$ and $C$, showing their intersections.
• Elements that are not in any of these sets ({C, S}) will be placed outside the circles but inside the universal set.

I will now generate a Venn diagram to illustrate this.{"prompt":"A Venn diagram illustrating the relationships among four sets: U, A, B, and C.\n\n- The universal set U is represented as a rectangle containing all distinct letters from the word 'MATHEMATICS': {M, A, T, H, E, I, C, S}.\n- Three overlapping circles represent sets A, B, and C.\n - Set A contains {M, A, T, H}.\n - Set B contains {A, I, M}.\n - Set C contains {T, E, A, M}.\n- The intersections of these sets are clearly shown:\n - A ∩ B = {M, A}\n - A ∩ C = {M, A, T}\n - B ∩ C = {M, A}\n - A ∩ B ∩ C = {M, A}\n- Elements {C, S} that do not belong to any of these sets are placed outside the circles but inside the universal set rectangle.","size":"1024x1024"}It looks like I couldn't generate the Venn diagram image automatically, but you can easily draw it using these steps:

How to Draw the Venn Diagram:

1. Draw a large rectangle – Label it as $U$ (Universal Set).
2. Draw three overlapping circles inside the rectangle, and label them as $A$, $B$, and $C$.
3. Place elements in the appropriate sections:
   • Center (A ∩ B ∩ C): Place {M, A} in the overlapping area of all three circles.
   • A ∩ B (but not C): No extra elements (already {M, A} in the center).
   • A ∩ C (but not B): {T}
   • B ∩ C (but not A): No extra elements (already {M, A} in the center).
   • A only: {H}
   • B only: {I}
   • C only: {E}
   • Outside all circles (but inside U): {C, S} (elements of U not in A, B, or C).

Would you like a step-by-step breakdown of how to label the diagram, or do you have any other questions?

Related Questions:

1. What are the possible subsets of $U$?
2. How do we find the union and intersection of more than two sets?
3. How do we determine if a set is a proper subset of another?
4. What is the cardinality of each set in this problem?
5. How does a Venn diagram help visualize set operations?

Tip:

When drawing Venn diagrams, always start with the innermost intersection (the smallest shared subset) and work outward to avoid missing elements.