A relation R from set A to set B is simply
an arbitrary subset of A × B.
If (a, b) ∈ R, we say "a is related to b", written a R b.
🌸Functions are just a special kind of relation!
🔑 Types of Relations in Set A
✏️ Empty Relation — no element related to any element: R = φ ⊂ A×A
🌍 Universal Relation — every element related to every element: R = A×A
🔁 Reflexive — (a, a) ∈ R ∀ a ∈ A
↔️ Symmetric — (a,b) ∈ R ⇒ (b,a) ∈ R
➡️ Transitive — (a,b)∈R & (b,c)∈R ⇒ (a,c)∈R
💡 Empty & Universal relations are called trivial relations.
⭐ Equivalence Relation
A relation that is Reflexive + Symmetric + Transitive all together!
Reflexive
+
Symmetric
+
Transitive
=
Equivalence Relation 🎀
Equivalence Class [a] = set of all elements related to a.
🧩 An equivalence relation partitions the whole set X into disjoint subsets (equivalence classes) whose union is X.
📝 Example: R = {(a,b): 2 divides a−b} on ℤ → splits ℤ into Evens [0] & Odds [1] — an equivalence relation!
✦ TYPES OF FUNCTIONS ✦
🧮 One-One, Onto & Bijective
Type
Meaning
Rule
One-One (Injective)
distinct inputs → distinct outputs
f(x₁)=f(x₂) ⇒ x₁=x₂
Onto (Surjective)
every element of Y is hit
Range(f) = Y
Bijective
One-One and Onto
f is invertible ✅
Many-One
2+ inputs → same output
opposite of one-one
🌟 Golden Rule (Finite sets only!): for f: X→X, one-one ⇔ onto. NOT true for infinite sets (e.g. f(x)=2x on ℕ is one-one but not onto).
^ each element paired uniquely — one-one AND onto!
🔗 Composition of Functions
If f: A→B and g: B→C, then:
A --f--> B
→
B --g--> C
gof(x) = g(f(x)) for all x ∈ A
⚠️ gof ≠ fog in general — order matters!
🔄 Invertible Functions
f: X→Y is invertible if ∃ g: Y→X such that:
gof = Iₓ and fog = I_Y g is called inverse of f, written f⁻¹.
💎 KEY THEOREM: f is invertible ⇔ f is one-one and onto (bijective)!
🗺️ Quick Mind-Map
Relation R ⊂ A×B
↓
Reflexive/Symmetric/Transitive
→
Equivalence Relation
Function f:X→Y
→
One-One / Onto
→
Bijective ⇒ Invertible
✅ Quick Revision Checklist
Define empty & universal relation
Reflexive, symmetric, transitive — definitions
Equivalence relation = R+S+T
Equivalence classes partition the set
One-one (injective) definition & test
Onto (surjective) definition & test
Bijective = one-one + onto
Finite set: one-one ⇔ onto (special property)
Composition gof(x) = g(f(x))
gof ≠ fog generally
Invertible ⇔ bijective
f⁻¹ satisfies gof=Iₓ, fog=I_Y
Practice: classify given relations (R,S,T)
Practice: check functions for 1-1 & onto
"There is no permanent place in the world for ugly mathematics." — G.H. Hardy 🌷