CDS Mathematics · Sets

Venn Diagrams and Counting

To count people in one group or another, add the groups and subtract the overlap once: |A ∪ B| = |A| + |B| − |A ∩ B|.

Why this matters

Twenty PYQs, three of them HARD, mostly in sets of three or four on one paragraph. Draw the Venn diagram and fill it from the INSIDE out: all three first, then each 'exactly two', then each 'only one'. Every question is then a sum of regions.

Concept 1 of 2: Two groups

Adding the two groups counts the people in both twice. Subtract the overlap once and you have everyone in at least one group; the rest of the total is in neither.

Definition

$|A \cup B| = |A| + |B| - |A \cap B|$ .
Neither $=$ total $-\ |A \cup B|$ . Only $A$ $= |A| - |A \cap B|$ .
'Failed in' figures: passed in both $= 100\% -$ failed in at least one.
If everyone is in at least one group, the overlap is $|A| + |B| -$ total.

Two sets

|A \cup B| = |A| + |B| - |A \cap B|

Worked example

40\%

failed in Maths,

30\%

in Science and

12\%

in both. What percentage passed in both?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2017 · CDS (II) 2017 — Elementary Mathematics · Q33Moderate

Example 1 · Sets · Venn Diagrams and Inclusion-Exclusion

In an examination, 35% students failed in Hindi, 45% students failed in English and 20% students failed in both the subjects. What is the percentage of students passing in both the subjects ?

Passed in both is not 100 minus failed in both

Those who passed both are everyone OUTSIDE the union of the 'failed' sets:

100 - (35 + 45 - 20) = 40\%

, not

100 - 20 = 80\%

Drill 6 more on two groups

Concept 2 of 2: Three groups

With three groups, fill the diagram from the centre. The pairwise figures usually include the centre, so subtract it to get each 'exactly two' region; then subtract those from each group to get 'only one'.

Definition

$|A \cup B \cup C| = |A| + |B| + |C| - |A \cap B| - |B \cap C| - |C \cap A| + |A \cap B \cap C|$ .
Exactly two $= \sum|A \cap B| - 3|A \cap B \cap C|$ .
Only $A$ $= |A| - |A \cap B| - |A \cap C| + |A \cap B \cap C|$ .
At least two $=$ exactly two $+$ all three. With three subjects, passing two or more $=$ failing at most one.

Three sets

|A \cup B \cup C| = \textstyle\sum|A| - \sum|A \cap B| + |A \cap B \cap C|

Worked example

Readers: paper I

20\%

, II

25\%

, III

15\%

; I and II

6\%

, II and III

5\%

, I and III

4\%

; all three

2\%

. What percentage read none?

Practice this conceptself-check · 4 quick reps

The same idea in a real exam question:

CDS · 2018 · CDS (II) 2018 — Elementary Mathematics · Q51Moderate

Example 2 · Sets · Venn Diagrams and Inclusion-Exclusion

Consider the following for the next 04 (four) items that follow : In an examination of Class XII, 55% students passed in Biology, 62% passed in Physics, 60% passed in Chemistry, 25% passed in Physics and Biology, 30% passed in Physics and Chemistry, 28% passed in Biology and Chemistry. Only 2% failed in all the subjects.

What percentage of students passed in all the three subjects ?

Pairwise figures include the centre

25\%

passed in Physics and Biology' counts the

4\%

who passed all three as well. Exactly Physics and Biology is

21\%

Drill 12 more on three groups

Summary — formulas & gotchas at a glance

A revision cheat-sheet for the formulas and gotchas above. Click any concept name to jump back to its full explanation.

Formulas (2)

Two groups
Two sets
$|A \cup B| = |A| + |B| - |A \cap B|$
Three groups
Three sets
$|A \cup B \cup C| = \textstyle\sum|A| - \sum|A \cap B| + |A \cap B \cap C|$

Watch out for (2)

Passed in both is not 100 minus failed in both
→ Two groups
Pairwise figures include the centre
→ Three groups

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