Set Theory

Set Theory is the mathematical study of collections of objects, known as sets. These objects (elements or members) could be numbers, alphabets, people, etc.

A set is a well-defined collection of distinct elements. For example:

• Set of natural numbers less than 5 = {1, 2, 3, 4}
• Set of vowels in English = {a, e, i, o, u}

In aptitude, Set Theory often appears in the form of Venn diagrams, counting elements in overlapping groups, and applying formulas to unions and intersections.

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2. Key Formulas & Shortcuts

Let:

• $n(A)$ = Number of elements in set A
• $n(B)$ = Number of elements in set B
• $n(A \cap B)$ = Elements common to A and B
• $n(A \cup B)$ = Total unique elements in A or B or both

Two Sets Formula

$$n(A \cup B) = n(A) + n(B) - n(A \cap B)$$

Three Sets Formula

$$n(A \cup B \cup C) = n(A) + n(B) + n(C) - n(A \cap B) - n(B \cap C) - n(C \cap A) + n(A \cap B \cap C)$$

Complement of a Set

If Universal Set = $n(U)$, and $n(A)$ are in A:

$$n(A') = n(U) - n(A)$$

Venn Diagram Tips

• Start from the innermost intersection and move outward.
• Subtract values from previous intersections to avoid double counting.

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3. Conceptual Tips & Common Mistakes

• Do not double count intersections; use the formulas strictly.
• Always begin with the highest overlap (all three sets) and subtract accordingly.
• Don’t assume mutual exclusivity unless specified.
• If “at least” or “only” is mentioned, pay close attention to Venn diagram placement.
• For “none” or “neither” type questions, calculate the union first, then subtract from total.

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5. Solved Examples

Example 1: Two Sets

Q: In a group of 100 students, 60 like Math, 45 like English, and 25 like both. How many like at least one subject?

A:

$$n(M \cup E) = 60 + 45 - 25 = 80$$

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Example 2: Two Sets with “Neither”

Q: In a group of 90 people, 50 drink tea, 40 drink coffee, and 20 drink both. How many drink neither?

A:

$$n(Tea \cup Coffee) = 50 + 40 - 20 = 70 \Rightarrow \text{Neither} = 90 - 70 = 20$$

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Example 3: Three Sets

Q: In a college of 200 students, 100 take Math, 80 take Physics, 60 take Chemistry; 30 take both Math and Physics, 20 take both Physics and Chemistry, 10 take both Math and Chemistry, and 5 take all three. How many students take at least one subject?

A:

$$n(M \cup P \cup C) = 100 + 80 + 60 - 30 - 20 - 10 + 5 = 185$$

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Example 4: Finding “Only”

Q: In a survey:

• 80 like A, 70 like B
• 50 like both A and B

How many like only A?

A:
Only A = 80 – 50 = 30

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Example 5: Universal Set and Complement

Q: Out of 120 people, 90 like cricket. How many don’t like cricket?

A:
Don’t like = $120 - 90 = 30$