2. Relations and Functions

7 topics2 h

Chapter 2 of NCERT Mathematics for Class 11: Relations and Functions. Three sections — Pairing the elements of two sets, Relations and Functions. Seven videos, 2 h in all.

Worked answers to the 36 exercise questions in this chapter: Exercise 2.1 · Exercise 2.2 · Exercise 2.3 · Miscellaneous Exercise

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Part 1

Pairing the elements of two sets

12 minWhy writing a pair in order carries information a set cannotA red shirt and a shirt, red: braces cannot tell them apart, since a set forgets order. Round brackets remember which came first — recovering fifteen distinctions braces lost.12 min · 13 parts13 minCounting all the pairs, and why swapping the two sets gives something elseTwo sets of size three and four cross into twelve pairs either way round — same count, almost never the same set. {1, 2, 3} and {2, 3, 4} share just four of those pairs.13 min · 14 parts

Part 2

Relations

14 minA relation is nothing more than a chosen part of the productThree letters, five names, fifteen possible pairings — one sentence keeps four of them. The relation is not the sentence. It is those four pairs, and nothing else.14 min · 14 parts14 minWhat goes in, what comes out, and what was merely allowed to come outFive pairs on the numbers one to six settle two sets outright — domain and range, read straight off the list. The third set, the codomain, was never on it.14 min · 13 parts

Part 3

Functions

16 minThe one-output rule that promotes a relation to a functionFive pairs on the numbers one to six, and no input sent to two places — yet it still fails to be a function, because six, the last number in the set, opens nothing.16 min · 14 parts19 minEach standard function is pinned down by its picture as much as by its ruleSeven named functions, and the thing that actually separates them is not what their formulas look like - it is how many expressions it took to write them down. Four need one. Three need cases. And every one of the three carries its case boundary in the picture, as a corner or as a jump, while not one of the four shows any such feature. That is measured here, function by function, not asserted.19 min · 15 parts15 minCombining two functions point by point, and the one case that failsAdd two functions, subtract them, scale one, multiply them: all four always work. Divide one by the other and it fails wherever the lower function is nothing.15 min · 13 parts

What you will be able to do

  • Explain why a two-element set cannot record which of its elements was chosen first, and produce a concrete pair of situations that the set notation confuses
  • Derive the size of a product from the construction rather than quoting it, and say why the construction produces no duplicates
  • State what Definition 2 identifies a relation with, and say what that identification rules out
  • Read a relation's domain, and its range, directly off a roster of pairs
  • Separate Definition 5 into its two demands and state each one as a test on a list of pairs
  • Name the seven functions §2.4.1 introduces and say, for each, whether one expression covers it or it is stated by cases
  • State how each of the five constructions in §2.4.2 defines the value of the new function at a given input

What this chapter assumes you already know

  • Set notation from Chapter 1: braces, membership with ∈, the empty set written φ
  • That a set is determined by which elements belong to it and by nothing else, so listing the same elements in a different order gives the same set
  • Subsets, intersection and union, at the level of naming which elements survive
  • Solving a pair of one-step linear equations in one unknown each
  • Plotting a point in the coordinate plane from its two coordinates
  • Intersection and union of sets, and the empty set
  • Counting subsets of a finite set: a set with n elements has 2ⁿ of them
  • Powers with small whole exponents, and reading 9 as 3 squared

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, b) is just another way of writing {a, b}."
  • "Two pairs are equal if they contain the same things."
  • "The product of two sets is a set of sets."
  • "φ has no elements, so crossing with it should leave the other set alone."
  • "Order only matters when the two sets are different."
  • "Fig 2.1 is an arrow diagram."
  • "A × B ≠ B × A means they are different sizes."
  • "× is multiplication, and multiplication commutes."