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CBSE Class 11 · Mathematics

Sets

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

This lesson follows the official chapter “Sets” 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
  • 9 concepts

1Sets and their Representations

A set is a well-defined collection: a test that lets you say in or out. Roster {2, 4, 6} and set-builder {x : x is an even whole from 2 to 6} name the same set. “Nice numbers” with no test is not a set.

A mood-list is not well-defined.

Figure. A set can be written two ways. Roster lists the members. Set-builder names the rule that decides membership. Both writings here name {2, 4, 6}.

How it works

  1. State the membership testThe definition.
  2. Write roster or set-builderThe representation.
  3. Refuse a fuzzy “nice” bagWell-defined.

2The Empty Set

The empty set has no members: {} or ∅. It is a set, not “nothing in the universe”. {∅} is a set whose one member is the empty set — different from ∅.

Writing {0} as empty is a miss — 0 is a member.

Figure. The empty set has no member you can point at. A set with 2 inside is not empty. Empty is a finished set, not a missing page.

How it works

  1. Ask whether any member passes the testNone → empty.
  2. Write ∅, not {0}The empty set.
  3. Keep {∅} as a one-element setA box holding empty is not empty.

3Finite and Infinite Sets

A finite set’s members can be counted to a last one; an infinite set has no last in that count. {1, 2, 3, 4} is finite; the even wholes are infinite. “Very many” is not the test — a last, or no last, is.

A long roster you got tired of is not proved infinite.

Figure. A finite set ends after its last member. An infinite set keeps offering a next member. The dashed arrow is the “and so on”, not a fourth listed number.

How it works

  1. Try to name a last memberFinite if you can, honestly.
  2. If the rule keeps making a next, call it infiniteNo last.
  3. Count n(A) only for finite AThis class.

4Equal Sets

Equal sets have the same members, order and repeats ignored: {2, 4, 4} = {4, 2}. Equal is a member-for-member test, not “same length of roster”.

{1, 2} and {1, 2, 2, 3} are not equal.

Figure. Equal sets have the same members, not merely the same count. A = {1, 3, 5} and B = {5, 1, 3} are equal. Writing them in a new order does not make a new set.

How it works

  1. List members without repeatsThe true collection.
  2. See if the other set has exactly thoseEqual.
  3. Ignore orderThe test.

5Subsets

A ⊆ B means every member of A is a member of B. A ⊂ B can mean proper (A ≠ B) if the lesson used that. ∅ ⊆ every set. A set is a subset of itself. Subset is not “smaller looking”.

Two members versus three is not automatically a subset.

Figure. A is a subset of B when every member of A is already a member of B. 3 sits in A, so it sits in B. 7 sits in B only, which is allowed for a proper subset.

How it works

  1. Take each member of AMust sit in B.
  2. If all do, write A ⊆ BThe subset.
  3. If also A ≠ B, it is proper as taughtProper.

A = {2, 4}, B = {2, 4, 6}

Is A a subset of B? Is B a subset of A?

  • 2, 4 in B?Yes
  • A ⊆ BYes
  • 6 in A?No — B ⊈ A

Pro tip. Every member of A must sit in B.

6Universal Set

A universal set U is the conversation’s whole bag for this problem — even wholes, or points in a plane, as named. Complements and Venn sketches live inside that U. A U you never named is a doodle frame.

Using “everything in the cosmos” as U is the wrong size.

Figure. The universal set U is the whole talk for this question. Every set named in the question is drawn inside U. A is one part of U, not a second universe.

How it works

  1. Name U for this problemThe conversation.
  2. Keep every set in the problem inside UHonest.
  3. Refuse a bigger U you did not needThis page.

7Venn Diagrams

A Venn diagram is a picture of membership: overlapping regions are “in both”. It is a mark-test, not decoration. Shading A ∩ B is the overlap; a pretty overlap with no members named is mute.

Two circles that do not match the membership test are a lying Venn.

Figure. A Venn diagram marks the same three membership cells — only A, A and B, only B — inside U. Circles cannot be drawn here, so the cells sit side by side. The middle cell is the overlap a circular Venn would show as a lens.

How it works

  1. Draw U as a box and the sets as regionsThe picture.
  2. Put members in the correct regionThe marks.
  3. Read overlap as “in both”Venn as a test.

8Operations on Sets

Union A ∪ B is “in A or B (or both)”; intersection A ∩ B is “in both”; difference A − B is “in A not B”. For A = {2, 4}, B = {4, 6}: union {2, 4, 6}, intersection {4}, A − B = {2}. Operations are member-tests, not new numbers.

Adding 2+4+6 as if union were a sum is a miss.

Figure. Union is every cell: only A, the shared cell, and only B. Intersection is the shared cell alone. Do not read the union bracket as if it named only the middle.

How it works

  1. Apply the or / and / not-in-B testThe operation.
  2. List the members that passThe result.
  3. Check with a Venn if you drew oneSame members.

9Complement of a Set

The complement A′ (in U) is “in U not in A”. If U = {1, 2, 3, 4, 5} and A = {2, 4}, A′ = {1, 3, 5}. Complement needs a named U. (A′)′ = A as a check.

Writing “not A” with no U is mute.

Figure. The complement of A inside U is whatever U still holds after A is taken out. “Not A” is a part of U, not a second universe outside the frame.

How it works

  1. Name U and AThe given.
  2. List U-members that fail AA′.
  3. Check (A′)′ = AThe opposite test.
{2, 4, 4} and {4, 2} are
  1. Equal sets — same members
  2. Different because of order
  3. Empty

Order and repeats do not change the set.

Notes

  • The official chapter title is “Sets”. 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

  • A ∪ B = {x : x∈A or x∈B}
  • A ∩ B = {x : x∈A and x∈B}
  • A′ = {x ∈ U : x ∉ A}

Recap

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

Sets and their Representations
A set is a well-defined collection: a test that lets you say in or out.
The Empty Set
The empty set has no members: {} or ∅. It is a set, not “nothing in the universe”. {∅} is a set whose one member is the empty set — different from ∅.
Finite and Infinite Sets
A finite set’s members can be counted to a last one; an infinite set has no last in that count.
Equal Sets
Equal sets have the same members, order and repeats ignored: {2, 4, 4} = {4, 2}.
Subsets
A ⊆ B means every member of A is a member of B.
Universal Set
A universal set U is the conversation’s whole bag for this problem — even wholes, or points in a plane, as named.

Practise Sets

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