Set theory

📚 Set Theory – Quick Study Notes for XAT (Quantitative Ability)

Why it matters
Set theory is the foundation of many XAT questions: counting problems, probability, logic, and even some “word” puzzles. Knowing the basics lets you tackle Venn diagrams, complement sets, and power‑set problems in seconds.

1. Simple Explanation

A set is a collection of distinct objects (called elements).

• Notation: { } (curly braces)
• Example: A = {1, 2, 3}

Key ideas:

• Subset – every element of one set is in another.
  B ⊆ A means B is a subset of A.
• Proper subset – subset but not equal.
  B ⊂ A means B is a proper subset of A.
• Union – all elements in either set.
  A ∪ B = {x | x ∈ A or x ∈ B}
• Intersection – only elements common to both.
  A ∩ B = {x | x ∈ A and x ∈ B}
• Complement – everything outside the set (relative to a universal set U).
  Aᶜ = U \ A
• Power Set – set of all subsets of a set.
  P(A) = {S | S ⊆ A}

De Morgan’s Laws (for complements):

• (A ∪ B)ᶜ = Aᶜ ∩ Bᶜ
• (A ∩ B)ᶜ = Aᶜ ∪ Bᶜ

2. Key Points (Bullet‑style Cheat Sheet)

Venn Diagram Tips

• Draw 2–3 overlapping circles (for 2–3 sets).
• Shade regions to represent “in” or “not in”.
• Helpful for visualizing unions, intersections, and complements.

3. Worked Example (XAT‑Style)

Problem
Let the universal set U = {1, 2, 3, 4, 5, 6, 7, 8}.
Define

• A = {1, 3, 5, 7}
• B = {2, 3, 6, 7}
• C = {4, 5, 6, 7}

Find the following:

1. |A ∪ B ∩ C|
2. (A ∩ B)ᶜ
3. P(A) (list all subsets of A)

Solution

Step 1 – Compute the intersection B ∩ C.
B ∩ C = {6, 7} (elements common to both).

Step 2 – Compute the union A ∪ (B ∩ C).
A ∪ {6,7} = {1,3,5,6,7}.
So |A ∪ B ∩ C| = 5.

Step 3 – Find the complement of (A ∩ B).
First, A ∩ B = {3, 7}.
Complement relative to U:
(A ∩ B)ᶜ = U \ {3,7} = {1,2,4,5,6,8}.

Step 4 – Power set of A.
A has 4 elements → |P(A)| = 2⁴ = 16 subsets.
List them (grouped by size):

Quick Summary of the Answer

• |A ∪ B ∩ C| = 5