1. Relations and Functions

8 topics3 h

Chapter 1 of NCERT Mathematics for Class 12: Relations and Functions. Three sections — Sorting relations by the properties they have, Sorting functions by how they map and Composing and undoing. Eight videos, 3 h in all.

Worked answers to the 35 exercise questions in this chapter: Exercise 1.1 · Exercise 1.2 · Miscellaneous Exercise

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Part 1

Sorting relations by the properties they have

16 minA relation as a subset of a product set, and the two extreme casesA relation stops meaning anything and becomes a set of pairs you chose - and that is the gain, not the loss. Measured rather than asserted: on a four-element set there are 65536 relations, exactly 1 of them sits inside every other and exactly 1 contains every other, and rules of the two shapes anyone actually writes down reach only 12 of the 65536.16 min · 13 parts17 minReflexive, symmetric and transitive as three demands that can fail independentlyThree demands are usually read as one checklist, and they are not: each asks about a different sized piece of the grid of pairs, and no one of them can force another. Measured rather than argued: of the 512 relations on a three-element set, every one of the 8 possible pass-and-fail patterns is occupied - 5 pass all three, 260 pass none - so not a single pattern is impossible.17 min · 13 parts18 minWhy an equivalence relation cuts its set into disjoint classes, and why the cut can be run backwardsReading “related to” as “in the same box” is a real claim, and it is false for most relations - because two boxes are either the same box or share nothing, and a relation is under no such obligation. Measured rather than argued: take each demand away in turn and the damage is different every time - 1 element in no box, then 2 boxes overlapping partly, then 6.18 min · 14 parts

Part 2

Sorting functions by how they map

20 minWhat one-one asks of distinct inputs, and what many-one allowsBeing one-one is a claim about collisions, and the same rule can have two different answers. Squaring the counting numbers from 1 to 200 offers 19 900 pairs and 0 collisions; the identical rule on the 201 whole numbers from -100 to 100 offers 20 100 pairs and 100 collisions. Same formula, character for character - only the domain changed.20 min · 14 parts22 minOnto as the demand that nothing in the codomain be left unhitOnto is not a property a rule has. It is a property a rule has with respect to two sets somebody chose. Doubling on the counting numbers leaves 20 of 40 declared targets unreached; the identical rule on the number line leaves none. And shrinking the DOMAIN can destroy coverage too - the reciprocal goes from 0 unreached to 18, with the target set never touched.22 min · 14 parts21 minBijections, and why on a finite set either half implies the other21 min · 13 parts

Part 3

Composing and undoing

18 minFeeding one function into another, and why swapping them changes the answer18 min · 11 parts20 minInvertibility as a two-sided undo, and why bijective is exactly the conditionUndoing a function is two equations, not one, and each of them buys exactly one property. Doubling on the counting numbers has a genuine undo on the input side - 40 inputs tried, 0 failures - and fails the output side at 20 of 40 targets. A folding function does the mirror image. Neither is invertible, and neither is a paradox.20 min · 11 parts

What you will be able to do

  • State the chapter's definition of a relation carried by one set, and apply it as a membership test on a stated pair
  • State each of the three conditions of Definition 3 and identify how many elements of A each one talks about at a time
  • State Definition 4 and verify all three demands for a relation defined by a divisibility condition
  • State Definition 5 in both its forms and explain why the second form is the one a proof uses
  • State Definition 6 and describe the two-step shape of a surjectivity proof: take an arbitrary element of the co-domain, produce an input that reaches it
  • State Definition 7 and check both halves separately for a stated function
  • State Definition 8 and identify the three sets involved and which two must match
  • State Definition 9, naming both equations and which set each identity function lives on

What this chapter assumes you already know

  • Ordered pairs, and the product of two sets written A × B (Class XI)
  • Reading and writing a set two ways — by listing its members, and by a rule
  • Subset, and that the empty set is a subset of every set
  • The absolute value of a real number, and that it is never negative
  • Divisibility and simple integer arithmetic
  • The number systems N, Z and R, and their standing symbols
  • Reading a relation from a rule as a list of ordered pairs on a finite set
  • Congruence of triangles, and perpendicular and parallel lines in a plane

Where people usually slip up

Sentences students actually say, taken from the notes the videos were made from. Each one is answered on the page of the video it belongs to.

  • "A relation has to mean something."
  • "The empty relation is a degenerate case, not really a relation."
  • "A rule that selects nothing must be a badly written rule."
  • "Universal relation means the two sets are the same size, or that A = B."
  • "An absolute value could come out negative if a is smaller than b."
  • "a R b and (a, b) belonging to R are two different facts."
  • "Relations are between two different sets, so a relation in one set is unusual."
  • "Symmetric plus transitive gives reflexive."