This lesson expands on what you learned in Grade 10, now focusing on problems that involve three sets. The key to success here is understanding how to represent the information in a Venn diagram and interpret the different regions correctly.

1. Understanding the Venn Diagram for Three Sets

A Venn diagram for three sets (A, B, and C) has 8 distinct regions. It is crucial to know what each region represents.

• Region 1: A ∩ B ∩ C - The very center. Belongs to all three sets (e.g., students who play all three sports).

• Region 2: (A ∩ B) ∩ C' - Belongs to A and B only.

• Region 3: (A ∩ C) ∩ B' - Belongs to A and C only.

• Region 4: (B ∩ C) ∩ A' - Belongs to B and C only.

• Region 5: A ∩ (B ∪ C)' - Belongs to A only.

• Region 6: B ∩ (A ∪ C)' - Belongs to B only.

• Region 7: C ∩ (A ∪ B)' - Belongs to C only.

• Region 8: (A ∪ B ∪ C)' - The area outside all circles. Belongs to none of the sets.

2. Key Vocabulary and Set Notation

3. Strategy for Solving Word Problems

Exam questions on sets are almost always word problems. Follow these steps to solve them:

1. Draw the Venn Diagram: Start by drawing a rectangle for the universal set (ε) and three overlapping circles for your sets (e.g., Cricket, Football, Volleyball).

2. Start from the Inside Out: This is the most important trick! Read the problem and find the number that belongs to all three sets (A ∩ B ∩ C). Write this number in the centremost region first.

3. Fill in the "Intersection of Two" Regions: The problem might say "15 students play Cricket and Football". This number (15) is for the entire overlapping area of C and F. Since you've already filled in the center (e.g., 6 students who play all three), you must subtract it to find the number who play "Cricket and Football only".

   • Cricket and Football only = 15 - 6 = 9. Write 9 in that region.

   • Repeat this for the other two "intersection of two" regions.

4. Fill in the "Only One" Regions: The problem might say "30 students play Cricket". This is the total for the entire Cricket circle. To find the number who play "Cricket only", subtract the three values you have already filled into the other parts of the Cricket circle.

5. Find the "None" Region: Add up all 7 numbers you have filled into the circles. Subtract this total from the universal set number (the total number of people surveyed) to find the number that goes outside the circles.

6. Answer the Questions: Once the Venn diagram is completely filled, you can answer any question by simply adding up the numbers in the relevant regions.

Exam Tips & Common Pitfalls

• Pitfall 1: Not starting from the center. If you don't fill in the A ∩ B ∩ C region first, all your other calculations will be wrong.

• Pitfall 2: Confusing "A and B" with "A and B only". "10 like Tea and Coffee" means the total overlap between the two circles. "10 like Tea and Coffee only" refers to the specific region that doesn't include the third category (Milk). The word "only" is extremely important.

• Exam Tip 1: Show your work on the diagram. A correctly filled-in Venn diagram will get you most of the marks, even if your final answer has a small calculation error.

• Exam Tip 2: Double-check your total. After filling all 8 regions, add all the numbers up. The sum must equal the total number of items in the universal set. If it doesn't, you've made a mistake somewhere.