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Venn Diagrams

Practice MCQs

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Reasoning Ability Venn Diagrams Competitive Exams

Venn Diagrams are visual reasoning tools used to show relationships between different groups or sets. They help us understand separate groups, overlapping groups, subsets, common elements, and numerical set problems.

What are Venn Diagrams?

A Venn Diagram represents groups using circles or closed shapes. If two groups have common elements, their circles overlap. If one group is completely inside another group, it means the smaller group is a part of the larger group.

In reasoning exams, Venn diagrams are used to test logical classification, group relations, common members, subset relations, and numerical calculations based on sets.

Quick idea: First decide whether the groups are separate, overlapping, or one group is completely inside another group.
Relationship Meaning Example
Separate Sets No common members Men and Women
Overlapping Sets Some members are common Teachers and Writers
Subset One group is fully inside another Dogs and Animals
Three-Set Overlap Three groups may share common members Students, Players, Singers

“Venn diagram questions become simple when the relationship between groups is identified first.”

Reasoning Tip
Key points
  • Circles represent sets or groups.
  • Overlap means common elements.
  • No overlap means no common members.
  • Inner circle means subset relation.
  • Read words like only, both, neither, and all carefully.
  • Fill common regions first in numerical questions.
sets overlap subset logic

Actual Venn Diagram Patterns

The following diagrams show the most common Venn diagram relationships used in reasoning questions.

1. Separate Sets
Men Women

No common members. Example: Men and Women in basic classification.

2. Overlapping Sets
Teachers Writers Common

Some teachers may be writers, and some writers may be teachers.

3. Subset Relation
Animals Dogs

All dogs are animals. So Dogs is completely inside Animals.

4. Three-Set Overlap
Students Players Singers All 3

Some persons may belong to one, two, or all three groups.

Note: These diagrams are inline SVG diagrams. They do not need JavaScript. If your sanitizer removes SVG, allow the tags svg, circle, and text with attributes like cx, cy, r, fill, stroke, viewBox, x, and y.

Important Venn Diagram Formulas

For two sets \(A\) and \(B\), the number of elements in at least one set is:

\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]

The number of elements only in \(A\) is:

\[ n(A \text{ only}) = n(A) - n(A \cap B) \]

The number of elements only in \(B\) is:

\[ n(B \text{ only}) = n(B) - n(A \cap B) \]

If the total number of people is known, then:

\[ \text{Neither} = \text{Total} - n(A \cup B) \]

Common Types of Venn Diagram Questions

Diagram Selection

Choose the correct diagram for the given relationship.

  • Dogs, Animals
  • Men, Women
  • Teachers, Writers
  • Roses, Flowers
Only Region

Find members belonging only to one group.

  • Only Hindi
  • Only Maths
  • Only Cricket
  • Subtract common part
Both / Common

Find elements present in two groups.

  • Both games
  • Both subjects
  • Common people
  • Intersection region
Neither Case

Find those who belong to none of the groups.

  • Use total
  • Find at least one first
  • Subtract from total
  • Useful in exams
Rule: In numerical Venn diagram problems, fill the common region first. Then calculate the only regions and finally calculate total or neither.

Step-by-Step Solving Method

Step Action Example
Step 1 Identify the sets or groups. Cricket and Football
Step 2 Check whether common members are possible. A student can like both games.
Step 3 Draw overlapping circles if common members are possible. Cricket circle overlaps Football circle.
Step 4 Fill the common region first. Both = 10 students.
Step 5 Use formula and calculate the required answer. \( n(A \cup B) = n(A) + n(B) - n(A \cap B) \)
Important: “Only A” means members in A but not in B. “Both” means members common to both A and B. “At least one” means members in A or B or both.

Worked Example with Actual Diagram

In a class, \(30\) students like Cricket, \(20\) students like Football, and \(10\) students like both. Find how many students like at least one game.

Cricket Football 20 10 10 Only Cricket Both Only Football
Cricket total = 30, Football total = 20, Both = 10. Therefore, only Cricket = 20 and only Football = 10.
\[ n(A \cup B) = n(A) + n(B) - n(A \cap B) \]
\[ n(A \cup B) = 30 + 20 - 10 = 40 \]

Therefore, 40 students like at least one game.

Solved Examples

Question Method Answer
Represent Dogs and Animals. All dogs are animals. So Dogs is a subset of Animals. Dogs inside Animals
Represent Teachers and Writers. Some teachers may be writers and some writers may be teachers. Overlapping circles
Represent Men and Women. In basic classification, these are separate groups. Separate circles
40 people like Tea, 30 like Coffee, and 15 like both. Find at least one. \[ n(A \cup B) = 40 + 30 - 15 = 55 \] 55
35 people know Hindi, 20 know English, and 10 know both. Find only Hindi. \[ n(A \text{ only}) = 35 - 10 = 25 \] 25
In a class of 70 students, 40 like Cricket, 35 like Football, and 20 like both. Find neither. \[ n(A \cup B) = 40 + 35 - 20 = 55 \] \[ \text{Neither} = 70 - 55 = 15 \] 15
Represent Squares, Rectangles, Circles. Squares are rectangles. Circles are separate from rectangles. Squares inside Rectangles; Circles separate
Represent Roses, Flowers, Animals. Roses are flowers. Animals are separate from flowers. Roses inside Flowers; Animals separate

Note: In two-set problems, always subtract the common part once because it is counted twice when adding both sets.

Common Traps and Shortcuts

Common Traps
  • Drawing overlap when no common member is possible.
  • Confusing “only A” with total A.
  • Adding common members twice.
  • Forgetting to subtract common part in union formula.
  • Ignoring “neither” in total-based questions.
  • Confusing subset relation with overlapping relation.
Useful Shortcuts
  • Use separate circles when groups cannot share members.
  • Use overlap when some common members are possible.
  • Use inner circle when one group is fully part of another.
  • Fill “both” first in numerical problems.
  • Use \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).
  • Use \(\text{Neither} = \text{Total} - n(A \cup B)\).
Exam approach: Identify whether the question is based on separate sets, overlapping sets, subset relation, only/common region, at least one, or neither.

Practice

A) Multiple Choice Questions
  1. Which diagram represents Dogs and Animals?
    Separate circles Overlapping circles Dogs inside Animals Animals inside Dogs
  2. Which diagram represents Teachers and Writers?
    Separate circles Overlapping circles Teachers inside Writers Writers inside Teachers
  3. Which diagram represents Men and Women?
    Separate circles Overlapping circles Men inside Women Women inside Men
  4. 40 students like Tea, 30 like Coffee, and 15 like both. How many like at least one?
    45 55 70 85
  5. 35 people know Hindi, 20 know English, and 10 know both. How many know only Hindi?
    15 20 25 30
B) Solve the Higher-Order Problems
  1. In a group of 60 students, 32 like Maths, 28 like Science, and 12 like both. How many like at least one subject? (Hint: Use \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).)
  2. In a class of 70 students, 40 like Cricket, 35 like Football, and 20 like both. How many like neither? (Hint: First find at least one.)
  3. Represent the relation: Roses, Flowers, Trees. (Hint: Roses are flowers; trees are separate.)
  4. Represent the relation: Squares, Rectangles, Circles. (Hint: Squares are rectangles; circles are separate.)
  5. 45 people read Newspaper A, 30 read Newspaper B, and 18 read both. How many read only Newspaper B? (Hint: Only B = B total - both.)
Reasoning Reminder

Venn diagram questions are based on understanding relationships between sets. Use separate circles for unrelated groups, overlapping circles for common members, and inner circles for subset relations.

Task: Create five Venn diagram questions using separate sets, overlapping sets, subset relation, two-set calculation, and neither-case calculation.

Show Suggested Answers
Multiple Choice
  1. Dogs inside Animals
    All dogs are animals. So Dogs is a subset of Animals.
  2. Overlapping circles
    Some teachers may be writers and some writers may be teachers.
  3. Separate circles
    Men and Women are treated as separate groups in basic Venn diagram classification.
  4. 55
    \[ n(A \cup B) = 40 + 30 - 15 = 55 \]
  5. 25
    \[ n(A \text{ only}) = 35 - 10 = 25 \]
Higher-Order Problems
  1. Maths = 32, Science = 28, Both = 12.
    \[ n(A \cup B) = 32 + 28 - 12 = 48 \] Answer = 48.
  2. Cricket = 40, Football = 35, Both = 20.
    \[ n(A \cup B) = 40 + 35 - 20 = 55 \] \[ \text{Neither} = 70 - 55 = 15 \] Answer = 15.
  3. Roses are flowers, so Roses circle is inside Flowers circle. Trees are separate from Flowers.
    Answer = Roses inside Flowers; Trees separate.
  4. Squares are rectangles, so Squares circle is inside Rectangles circle. Circles are separate from Rectangles.
    Answer = Squares inside Rectangles; Circles separate.
  5. Newspaper B = 30, Both = 18.
    \[ n(B \text{ only}) = 30 - 18 = 12 \] Answer = 12.
Clue Explanation

In Venn diagram numerical problems, the common region is counted in both sets. Therefore, subtract the common part once while finding the union.

Exam tips
  • Identify whether groups are separate, overlapping, or subsets.
  • Read only, both, neither, and at least one carefully.
  • Fill common region first in numerical questions.
  • Use \(n(A \cup B) = n(A) + n(B) - n(A \cap B)\).
  • Use \(\text{Neither} = \text{Total} - n(A \cup B)\).
  • Do not assume overlap unless common members are possible.
Practice MCQs