Are these Relations and Functions notes free?

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What does the Relations and Functions revision sheet cover?

Relation, Equivalence relation, Equivalence classes, Empty relation vs Universal relation, Empty relation, Universal relation, Reflexive, symmetric, transitive: defined, a always sits in its own class, b belongs to a's class exactly when they're related, Quick relation ID card, a ≤ b, on R, a is parallel to b, on lines.

Is this for Class 12 CBSE Mathematics?

It is written from the NCERT syllabus, so it serves CBSE and any board that follows NCERT for Mathematics.

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Relations and Functions — Class 12 CBSE Mathematics revision notes

A 4-page visual revision sheet for Relations and Functions, written for Class 12 CBSE Mathematics. Formulas, diagrams and the traps that cost marks, laid out to be read the night before rather than the term before. Free, and it opens inside CapCompass.

Relations and Functions revision notes, page 1 of 4 — A preview of page 1 — all 4 pages open full size in the app.

Relations and Functions in one look

which pairings survive being reflexive, symmetric, transitive, and which functions can be run backward

Relation: any subset of ordered pairs from A to A
Equivalence relation: reflexive, symmetric AND transitive, all at once
Equivalence classes: A splits into disjoint pieces, every element in exactly one class

Empty relation vs Universal relation

Empty relation

R contains no pairs at all, R = ∅
reflexive only if A itself is empty, otherwise it fails immediately
vacuously symmetric and transitive, there's nothing to violate

Universal relation

every pair (a,b) is in R, R = A × A
reflexive, symmetric, transitive all hold trivially
always an equivalence relation, since the whole set collapses to one class

Reflexive, symmetric, transitive: defined

reflexive: (a,a) ∈ R for every a ∈ A, no element skipped
symmetric: (a,b) ∈ R ⇒ (b,a) ∈ R, the pair runs both ways
transitive: (a,b) ∈ R and (b,c) ∈ R ⇒ (a,c) ∈ R, chains close up
all three together, and only then, R earns the name equivalence relation

Equivalence classes

a always sits in its own class a in [a]
b belongs to a's class exactly when they're related bin[a] ↔ aRb

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What this Relations and Functions sheet covers

Relation
Equivalence relation
Equivalence classes
Empty relation vs Universal relation
Empty relation
Universal relation
Reflexive, symmetric, transitive: defined
a always sits in its own class
b belongs to a's class exactly when they're related
Quick relation ID card
a ≤ b, on R
a is parallel to b, on lines
a is perpendicular to b, on lines
a divides b, on positive integers
f
g
g∘f
Composing right to left
One-one vs Many-one
One-one (injective)
Many-one
Is f(x) = x squared one-one?
Invertible functions
inverse undoes f both ways
domain and range trade places
Onto vs Into
Onto (surjective)
Into
Bijective = one-one AND onto
Why bijective, piece by piece
Undoing two steps in a row
if f and g are both invertible
Traps that cost marks
Tricks that save time

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