Deepening your understanding — equivalence relations, bijective functions, and binary operations
"There is no permanent place in the world for ugly mathematics." — G.H. Hardy
In Class XI, you learned the basics of relations and functions. Now we'll explore deeper properties that make certain relations special.
Every element is related to itself: (a, a) ∈ R for all a in A
If a relates to b, then b relates to a: (a,b) ∈ R → (b,a) ∈ R
If a→b and b→c, then a→c: (a,b)∈R and (b,c)∈R → (a,c)∈R
Reflexive + Symmetric + Transitive = Equivalence Relation
An equivalence relation divides a set into non-overlapping groups called equivalence classes. Every element belongs to exactly one class.
Example: "Same parity" on integers creates two classes: {even numbers} and {odd numbers}.
Functions can be classified by how they map elements:
Different inputs → different outputs. No two inputs share the same output.
Every element in the codomain is hit by some input. Range = Codomain.
Both one-one AND onto. Perfect pairing — every input has a unique output, and every output is used.
Multiple inputs can share the same output. Not injective.
For finite sets, a function f: X → X is one-one if and only if it's onto. This is a special property of finite sets — it doesn't hold for infinite sets!
You can chain functions together — the output of one becomes the input of another.
Given f: A → B and g: B → C, the composition g∘f: A → C is defined by:
(g∘f)(x) = g(f(x))
Apply f first, then g to the result. Order matters!
g∘f ≠ f∘g in general!
Example: f(x) = cos x, g(x) = 3x²
• (g∘f)(x) = 3cos²x
• (f∘g)(x) = cos(3x²)
These are completely different functions!
One-One: If f and g are both one-one, then g∘f is one-one
Onto: If f and g are both onto, then g∘f is onto
Associative: h∘(g∘f) = (h∘g)∘f
An invertible function can be "undone" — there exists a reverse function that takes you back.
f: X → Y is invertible if there exists g: Y → X such that:
g∘f = Iₓ (identity on X)
f∘g = Iᵧ (identity on Y)
The inverse g is denoted f⁻¹.
A function is invertible if and only if it's bijective (one-one AND onto).
• Not one-one? → Can't reverse (multiple inputs map to same output)
• Not onto? → Some outputs have no input to reverse to
(g∘f)⁻¹ = f⁻¹ ∘ g⁻¹
To undo a composition, reverse the order and invert each function. Like taking off shoes and socks — you remove shoes first (last thing put on), then socks.
A binary operation takes two elements from a set and produces another element in the same set.
A binary operation ∗ on set A is a function: A × A → A
It takes two elements from A and gives back one element in A.
Examples: +, ×, −, ÷ (on appropriate sets), ∪, ∩ (on sets)
a ∗ b = b ∗ a for all a, b. Order doesn't matter.
(a ∗ b) ∗ c = a ∗ (b ∗ c). Grouping doesn't matter.
Element e where a ∗ e = a = e ∗ a for all a.
For element a, inverse a⁻¹ where a ∗ a⁻¹ = e.
• Subtraction: NOT commutative (3−5 ≠ 5−3), NOT associative
• Division: NOT commutative, NOT associative
• Addition: Commutative, associative, identity = 0
• Multiplication: Commutative, associative, identity = 1
• Equivalence relation: Reflexive + Symmetric + Transitive → partitions set into equivalence classes
• One-One (Injective): Different inputs → different outputs
• Onto (Surjective): Every output is hit by some input
• Bijective: One-one + onto = invertible
• Composition: (g∘f)(x) = g(f(x)). Order matters!
• Inverse: (g∘f)⁻¹ = f⁻¹ ∘ g⁻¹
• Binary operation: A × A → A. Can be commutative, associative, have identity/inverse