Venn Diagrams
Venn Diagrams are a visual reasoning topic of the ADRE Grade IV paper in which the relationship between two or more classes is shown with the help of circles. The candidate may be asked to choose the diagram that best represents a given group of items, or to say which items a shown diagram represents, or to count how many persons belong to a particular region. The topic is close to set theory and is best solved by understanding the four basic ways in which two circles can be placed.
Meaning of a Venn Diagram
A Venn diagram uses circles inside a rectangle to represent classes and their relationships. Each circle stands for one class, and the rectangle stands for the whole universe. The way the circles overlap tells whether the classes share members or not. The same idea is used in everyday questions such as "Which diagram best shows Doctors, Men and Fathers?" where the three classes partly overlap. Reading the picture correctly is the whole skill of this topic.
Four Basic Relationships
- Fully overlapping — one class is completely inside another, as in "All roses are flowers".
- Partly overlapping — the classes share some members, as in "Some doctors are men".
- Separate or disjoint — the classes share no member, as in "Cats and dogs".
- Three circles — one may be inside another, or all three partly overlap, depending on the items.
Common Examples
Mothers, Women and Teachers are three classes in which all mothers are women, so the mother circle lies inside the women circle, while some women are teachers, so the teacher circle overlaps the women circle. In Birds, Parrots and Fish, all parrots are birds and no bird is a fish, so the parrot circle lies inside the bird circle and the fish circle is separate. In Flowers, Roses and Lilies, both roses and lilies are flowers, so two separate circles lie inside the flower circle.
Counting Regions
Some questions give the number of persons in each class and its overlaps and ask for the number in a particular region. The safest method is to start filling the innermost region where all the circles overlap, then move outward to the two-circle overlaps, and finally fill the single circles. Always subtract the already-filled numbers. If the total is given, the number outside the circles can also be found by subtracting the sum of all filled regions from the total.
Worked Example
Suppose among thirty students, fifteen play cricket, twelve play football and five play both. The number who play only cricket is 15 − 5 = 10 and only football is 12 − 5 = 7. The number who play at least one game is 10 + 5 + 7 = 22, so 30 − 22 = 8 students play neither game. In a relationship question, "Doctors, Men, Fathers" is best shown by three circles that partly overlap, because a doctor may be a man, a man may be a father, and a father may be a doctor, but none is fully inside another.
How This Topic Appears in the Examination
Venn diagram questions usually appear as two to four objective questions of one mark each. They may be relationship questions asking for the best diagram, or counting questions with given numbers. The relationship type is answered by matching the statement to one of the four basic patterns, and the counting type is answered by careful addition and subtraction. The questions are quick when the circles are drawn neatly.
Memory Tips
Decide first whether the classes are independent, partly related or fully related, because this fixes the pattern. For "All A are B" put A inside B; for "Some A are B" make the circles overlap; for "No A is B" keep them separate. In counting questions, always begin from the innermost overlap and work outwards. Draw the circles roughly but keep the overlapping portions large enough to write numbers inside.
Common Mistakes to Avoid
- Putting a class inside another when the statement says only "some".
- Overlapping circles when the statement says "no".
- Forgetting to subtract the overlap while filling the single circles.
- Ignoring the part outside the circles in counting questions.
- Matching the diagram by guess without checking every statement.
How to Prepare This Topic
- Learn the four basic circle patterns by heart.
- Practise converting statements into diagrams.
- Solve counting questions starting from the innermost region.
- Check the answer by adding all regions back to the total.
- Practise both relationship and counting questions daily.
More Solved Examples
In a survey of fifty people, thirty like tea, twenty-five like coffee and fifteen like both. The number who like only tea is 30 − 15 = 15, only coffee is 25 − 15 = 10, and at least one drink is 15 + 15 + 10 = 40, so ten people like neither. For a relationship question, "Teachers, Fathers, Men" is shown by three circles that partly overlap, because a teacher may be a father, a father is usually a man, and a teacher may be a man, but none of the classes lies fully inside another.
Advanced Tips for the Exam
For relationship questions, check the pattern that matches every part of the statement, not just the first word. For counting questions, always begin at the centre where all the circles overlap, and subtract as you move outward. If three classes are given, fill the triple overlap first, then the three double overlaps, and finally the single circles. Draw the circles large enough to hold the numbers inside them. Always verify by adding all the regions and comparing the sum with the given total.
Quick Revision Checklist
- Four basic patterns: inside, overlap, separate and triple.
- "All A are B" means A lies inside B.
- "Some A are B" means the circles overlap.
- "No A is B" means the circles are separate.
- Counting starts from the innermost region.
- Total = sum of all regions plus the outside part.
Frequently Asked Questions
What is a Venn diagram?
A Venn diagram uses circles inside a rectangle to show the relationship between classes, such as inside, overlapping or separate.
How do I show "All A are B" in a Venn diagram?
Draw the circle of A completely inside the circle of B, because every member of A is also a member of B.
How do I solve counting questions with Venn diagrams?
Start with the innermost overlap where all circles meet, then fill the two-circle overlaps by subtracting, then the single circles, and finally the part outside the circles.
How can I check my answer in a counting question?
Add up all the regions you have filled and see whether the total matches the number given in the question. The difference, if any, is the part outside the circles.
Are Venn diagrams related to syllogism?
Yes. Both use circles to show relationships between classes, and the same diagram can often be used to test conclusions in syllogism questions.