Set Applications with Venn Diagrams

Venn diagrams help you visualise how sets relate to each other. They are especially useful when solving word problems involving union, intersection, complement, and number of elements in sets.

 

Venn Diagrams: Key Regions

In a two-set diagram (sets A and B), the regions are:

• $A \cap B$: The overlapping region (common to both A and B)
• $A \cup B$: Everything inside A, B, or both
• $A'$: Everything outside A (in the universal set)

In a three-set diagram (A, B, C), there are 8 regions representing all combinations of overlap and exclusion.

 

Solving Word Problems with Venn Diagrams

These steps help when solving real-life set problems using Venn diagrams:

1. Draw a rectangle for the universal set and circles for each set involved.
2. Label the diagram clearly.
3. Start by filling the most specific regions first — usually the intersection(s).
4. Use subtraction to fill outer regions.
5. Add up all the values and compare to the total in the universal set to find missing values.

 

Worked Example 1 (Two Sets)

 

Worked Example 2 (Three Sets)

 

Symbols to Remember

• $A \cup B$: Union (either A or B or both)
• $A \cap B$: Intersection (both A and B)
• $A'$: Complement (everything not in A)
• $n(A)$: Number of elements in A