Week 1 Notes: Sets, Relations, and Functions

Course: Jan 2026 - Mathematics I Difficulty: Foundational Focus: Abstract structures and mappings

1. Set Theory Fundamentals

1.1 Types of Sets

• Empty Set ($\emptyset$): Contains no elements. $|Ø| = 0$.
• Singleton Set: Contains exactly one element.
• Finite vs. Infinite: Based on whether the count of elements terminates.
• Power Set $P(A)$: The set of all subsets. If $|A| = n$, then $|P(A)| = 2^n$.

1.2 Operations

2. Relations

2.1 Properties of Relations (on Set A)

1. Reflexive: $\forall a \in A, (a, a) \in R$.
2. Symmetric: If $(a, b) \in R$, then $(b, a) \in R$.
3. Transitive: If $(a, b) \in R$ and $(b, c) \in R$, then $(a, c) \in R$.

3. Functions

3.1 Key Classifications

• Injective (One-to-One): No two distinct elements in $A$ have the same image in $B$.
  • Check: $f(x_1) = f(x_2) \implies x_1 = x_2$.
• Surjective (Onto): Range = Co-domain. Every element in $B$ has at least one pre-image in $A$.
• Bijective: Both Injective and Surjective. Bijective functions have inverses.

4. Common Traps & Do's/Don'ts

❌ Don'ts (Common Mistakes)

• Miscounting Subsets: Forgotten that the empty set $\emptyset$ is a subset of every set.
• Mistaking $\in$ vs $\subset$:
  • $a \in \{a, b\}$ is True.
  • $\{a\} \subset \{a, b\}$ is True.
  • $\{a\} \in \{a, b\}$ is False.
• Onto Confusion: Thinking a function is onto just because it's defined. Always check if the Co-domain is fully covered.

✅ Do's

• Use Venn Diagrams: Always sketch logic for 3-set problems.
• Horizontal Line Test: Use it to check for Injectivity on graphs.
• Vertical Line Test: Use it to verify if a relation is even a function.

5. Mnemonics & Quick Recall

• PIE (Principle of Inclusion-Exclusion): $|A \cup B \cup C| = (|A|+|B|+|C|) - (|A \cap B|+|B \cap C|+|C \cap A|) + |A \cap B \cap C|$ Think: Add all singles, Subtract all doubles, Add the triple.
• Relation Acronym: RST
  • Reflexive (Selfie)
  • Symmetric (Mirror)
  • Transitive (Bridge)

6. Representative Examples

Example: Set Cardinality

• Elements of $A$ are: $1$, $2$, and the set $\{3, 4\}$.
• Count $n = 3$.
• $|P(A)| = 2^3 = 8$.