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CBSE Class 11 · Mathematics

Relations and Functions

Official NCERT chapter from Mathematics (book code kemh1). ExamMaster notes are original teaching at CBSE Class 11 depth.

This lesson follows the official chapter “Relations and Functions” in Mathematics. The words below are ExamMaster’s teaching, not a paste from the book. Use the NCERT chapter for the classroom sequence; use these notes to hold the idea without copying exercises or figures.

  • CBSE Class 11
  • Medium level
  • 8 concepts

1Cartesian Product of Sets

The Cartesian product A × B is the set of ordered pairs (a, b) with a from A and b from B. Order inside the pair matters: (2, 5) is not (5, 2). If A = {1, 2} and B = {5, 7}, A × B has four pairs. A bag of unpaired members is not this product.

Writing A ∪ B as if it were A × B is the heading-steal.

Figure. A × B is the set of every ordered pair (a, b) with a from A and b from B. Here A = {1, 2} and B = {a, b} give four pairs, not a Venn overlap.

How it works

  1. Name A and BThe two sets.
  2. List every (a, b)The product.
  3. Keep order inside each pairCartesian.

A = {1, 2}, B = {5, 7}

List A × B.

  • Pairs(1,5), (1,7), (2,5), (2,7)
  • Count2 × 2 = 4
  • Not(5,1) is B × A

Pro tip. Ordered pairs; n(A×B) = n(A)n(B) when finite.

2Relations

A relation from A to B is a subset of A × B: some pairs you keep, some you drop. “a is less than b” on {1, 2} × {5, 7} keeps (1,5), (1,7), (2,5), (2,7) if all hold, or a tighter rule if you name one. A relation is a chosen subset, not a feeling about the sets.

Saying “they are related” with no pair-list is a heading.

Figure. A relation is any set of arrows from A to B. Element 1 may send two arrows; element 2 may send none. That is still a relation.

How it works

  1. Form A × BThe possible pairs.
  2. Keep the pairs the rule namesThe relation.
  3. Write it as a set of pairsA subset.

3Functions

A function f : A → B assigns to each a in A exactly one b in B. Every input used once; no input with two outputs. A = {1, 2}, f(1)=5, f(2)=5 is allowed (two-to-one); f that sends 1 to both 5 and 7 is not a function.

A pair-list that misses an input is not this function.

Figure. A function from A to B sends each element of A to exactly one element of B. Two inputs may share an image; one input may not fork.

How it works

  1. Name domain A and codomain BThe two sets.
  2. Give each a exactly one bThe assign.
  3. Refuse a two-output or a missed inputThe function-test.

4A definition is a test you can run

Relation / function is a test: a subset of A × B, plus “exactly one” if you claim a function. If you only say “maps”, you have a heading.

A pretty arrow-sketch with a leftover input is not the test.

Figure. The function definition is a test: for the x you picked, count images in B. Exactly one passes. Zero or two-or-more fails.

How it works

  1. Name A, B, and the pair-listThe object.
  2. Ask subset, then exactly-oneThe test.
  3. Then the word has contentThe definition ran.

5Name the given before the unknown

The given are A, B, and the rule. The unknown is the pair-list or “is it a function?”. Copy A before you invent an extra input.

Using B × A because the picture was drawn backwards is a silent given.

Figure. Write the given sets A and B before you hunt for the rule. The unknown is the pairing, not the names of the sets.

How it works

  1. Copy A, B, and the ruleThe given.
  2. Name the product, relation, or function-verdictThe unknown.
  3. Then list pairsGiven first.

6One worked case is enough at this class

One product or one function-check is enough. A page of the same four-pair clone does not add a new exactly-one.

Ten twins of (1,5) are still one idea.

Figure. One fully written map is enough to run the test: each of 1, 2, 3 has one image. Range is {a, b}; b is hit once, a twice.

How it works

  1. Run one A × B or one f-checkThe case.
  2. A second only as a checkOptional.
  3. StopThis class.

7A miss: swapping the school name for the picture

A seating-chart photo is a setting. The maths is A × B and a pair-rule. A smile is not an ordered pair.

A family-tree still is not a function until each input has one output.

Figure. The chapter name is not the relation. The picture is the pairing of elements. Do not treat the title as if it were the map.

How it works

  1. Keep the chart as a storyA setting.
  2. Write A, B, and the pairsThe maths.
  3. Then the exactly-one testDo not swap.

8Check by the opposite action or the opposite test

Check a function by looking for a missed input or a double output. Check A × B by counting n(A)n(B). Four pairs from two-by-two must match the list.

A Venn of A and B is not the opposite test for a product of pairs.

Figure. Check a claimed function by the opposite walk: from an image, name its preimage. Here a returns to 1 and b returns to 2.

How it works

  1. State the four pairs or “f is a function”The claim.
  2. Recount or hunt a double outputThe check.
  3. Mend if they disagreeHonest.
A rule that sends 1 to both 5 and 7
  1. Is not a function — two outputs for one input
  2. Is a function because 5 and 7 are in B
  3. Is A ∪ B

Exactly one b for each a.

Notes

  • The official chapter title is “Relations and Functions”. Teach the school test for that title, not a contest shortcut.
  • If a step needs a later class, stop. The next official chapter will pick it up.

Formulas

  • n(A×B)=n(A)n(B) when finite
  • function: each a in A has exactly one f(a) in B

Recap

Hold these pegs from the official chapter “Relations and Functions”. The wording is ExamMaster’s teaching, not a textbook recap.

Cartesian Product of Sets
The Cartesian product A × B is the set of ordered pairs (a, b) with a from A and b from B.
Relations
A relation from A to B is a subset of A × B: some pairs you keep, some you drop.
Functions
A function f : A → B assigns to each a in A exactly one b in B.
A definition is a test you can run
Relation / function is a test: a subset of A × B, plus “exactly one” if you claim a function.
Name the given before the unknown
The given are A, B, and the rule. The unknown is the pair-list or “is it a function?”. Copy A before you invent an extra input.
One worked case is enough at this class
One product or one function-check is enough.

Practise Relations and Functions

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  • 1 quick check with worked explanations
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