Q.If $A$ and $B$ are two sets such that $A \subset B$, what are $A \cup B$ and $A \cap B$? Justify your answer with the help of a Venn diagram description.

Venn Diagrams: Seeing Sets as Pictures

A Venn diagram is a picture of how groups of things overlap. Before you learn the formal definition, think of it this way: you have a circle for "things that are red" and another circle for "things that are fruits." Some things — like an apple — belong in both circles. The overlapping region is where the apple lives. That's the entire idea.

The Intuition

Imagine you have two collections:

• Set A: Students who play cricket.
• Set B: Students who play football.

Some students play only cricket, some only football, some play both, and some play neither. A Venn diagram draws each set as a circle (or oval) inside a rectangle. The rectangle is the "universe" — all the students you're considering. The circles overlap where students belong to both sets.

The picture instantly shows you:

• The intersection (both sports) — the overlapping lens.
• The union (at least one sport) — everything inside either circle.
• The complement (neither sport) — the part of the rectangle outside both circles.

The Precise Statement

A Venn diagram is a graphical representation of sets and their relationships, using closed curves (usually circles or ellipses) drawn inside a rectangle that represents the universal set $U$. Each curve encloses the elements of one set. The spatial arrangement — especially overlaps — encodes logical relationships.

For two sets $A$ and $B$, the diagram divides the rectangle into four disjoint regions:

For three sets, you use three overlapping circles, creating eight regions (including the outside). The principle is the same: every possible combination of membership is represented by a distinct region.

Why They Work

The power of a Venn diagram is that it turns a logical problem into a visual one. Instead of manipulating symbols like $A \cap (B \cup C)$, you shade regions and see the answer directly. This is especially useful for:

• Checking set identities (e.g., De Morgan's laws).
• Solving word problems with "and," "or," "not."
• Counting problems where you need to find how many elements are in each region.

A Simple Example

Problem: In a class of 30 students, 18 like mathematics, 15 like physics, and 10 like both. How many like neither? …