Subsets
Boys of your class are part of the whole class. Every part is a subset of the whole.
A is a subset of B (written A ⊆ B) if every element of A is also in B. If A ⊆ B and A ≠ B, then A is a proper subset (A ⊂ B). The empty set is a subset of every set, and every set is a subset of itself.
Rule 1: Check A ⊆ B — Pick each element of A and look for it in B.
- Easy: {1, 2} ⊆ {1, 2, 3}: yes.
- Medium: {2, 5} ⊆ {1, 2, 3}: no, 5 is missing.
- Exam-type: N ⊂ Z ⊂ Q ⊂ R: each is a proper subset of the next.
Rule 2: Proper subset — A ⊂ B means A ⊆ B but B has something more.
- Easy: {a} ⊂ {a, b}: proper.
- Medium: {1, 2, 3} ⊆ {3, 2, 1} but it is not proper (they are equal).
- Exam-type: ∅ ⊂ {5}: proper subset.
A = B exactly when A ⊆ B and B ⊆ A.
Mixing ∈ and ⊆. Use ∈ between an element and a set, ⊆ between two sets: 2 ∈ {1, 2} but {2} ⊆ {1, 2}.
∈ is for a member, ⊆ is for a group.
Spot the difference
| ∈ | ⊆ |
|---|---|
| Element and set | Set and set |
| 3 ∈ {1, 3} | {3} ⊆ {1, 3} |
Practice questions
- 1Is {3} ⊆ {1, 2, 3}?
- 2Is 3 ⊆ {1, 2, 3} a correct statement?
- 3Is ∅ a subset of {7}?
- 4Is the set of squares of natural numbers a subset of N?
- 5If A ⊆ B and B ⊆ A, what can you say?
- 6Is {x : x ∈ N, x is a multiple of 6} ⊆ {x : x ∈ N, x is a multiple of 3}?
- 7Is the set of multiples of 3 a subset of the multiples of 6?
- 8A = {1, 2}. Write all subsets of A.
Show answers (try them yourself first)
- 1Yes.
- 2No — 3 is an element, write 3 ∈ {1, 2, 3}.
- 3Yes — ∅ is a subset of every set.
- 4Yes.
- 5A = B
- 6Yes — every multiple of 6 is a multiple of 3.
- 7No — 3 is a multiple of 3 but not of 6.
- 8∅, {1}, {2}, {1, 2}
Number of subsets
For each of your 3 friends you decide: invite or not. That gives 2 choices each time and 2 × 2 × 2 = 8 possible guest lists.
A set with n elements has 2ⁿ subsets, of which 2ⁿ − 1 are proper subsets (all except the set itself). The set of all subsets of A is the power set P(A).
Formula: Number of subsets = 2ⁿ
Rule 1: Count subsets — Count the elements n first, then compute 2ⁿ.
- Easy: n = 2: 2² = 4 subsets.
- Medium: {a, b, c, d}: 2⁴ = 16 subsets.
- Exam-type: A = {x : x ∈ N, x ≤ 6}: n = 6, subsets = 64, proper subsets = 63.
Rule 2: Power set — List subsets by size: 0 elements, 1 element, 2 elements, ….
- Easy: P({1}) = {∅, {1}}.
- Medium: P({1, 2}) has 4 members: ∅, {1}, {2}, {1, 2}.
- Exam-type: If n(A) = 3, then n(P(A)) = 8.
The empty set and the set itself are always counted among the 2ⁿ subsets.
Writing 2n instead of 2ⁿ.
Each extra element doubles the number of subsets.
Add one element — double the subsets.
Practice questions
- 1How many subsets does a set of 4 elements have?
- 2How many proper subsets does {p, q} have?
- 3Write P({x}).
- 4A set has 32 subsets. Find n.
- 5How many subsets of {1, 2, 3, 4, 5} have exactly 1 element?
- 6How many subsets of {1, 2, 3, 4} contain the element 1?
- 7How many subsets of {a, b, c} have exactly 2 elements?
- 8If n(A) = 7, how many non-empty proper subsets does A have?
Show answers (try them yourself first)
- 116
- 22² − 1 = 3
- 3{∅, {x}}
- 42⁵ = 32, so n = 5
- 55
- 6Choose freely among the other 3 elements: 2³ = 8
- 73
- 82⁷ − 2 = 126
Intervals as subsets of R
Bus tickets are valid "from 6 a.m. to 9 a.m." — a stretch on the time line. Stretches of the number line are called intervals.
For real numbers a < b: the open interval (a, b) = {x : a < x < b} leaves out both ends; the closed interval [a, b] = {x : a ≤ x ≤ b} includes both; [a, b) and (a, b] include one end. The length of any such interval is b − a.
In the figure: The interval (1, 5]: open circle at 1 (not included), filled circle at 5 (included).
Rule 1: Read an interval — Round bracket = end left out, square bracket = end included.
- Easy: (2, 7) = {x : 2 < x < 7}.
- Medium: [−1, 3) = {x : −1 ≤ x < 3}.
- Exam-type: (−∞, 4] = {x : x ≤ 4}; ∞ always gets a round bracket.
Rule 2: Write as an interval — Decide for each end whether it is included.
- Easy: {x : 0 ≤ x ≤ 10} = [0, 10].
- Medium: {x : −5 < x ≤ 2} = (−5, 2].
- Exam-type: {x : x > 3} = (3, ∞).
Length of [2, 9] and of (2, 9) is the same, 7.
Using a square bracket with ∞.
Round = Reject the end, Square = Seat the end.
Spot the difference
| (a, b) | [a, b] |
|---|---|
| Ends not included | Ends included |
| Open circle ○ | Filled circle ● |
Practice questions
- 1Write {x : 1 < x < 8} as an interval.
- 2Write [−2, 5) in set-builder form.
- 3Find the length of the interval [3, 11].
- 4Is 5 in the interval (5, 9]?
- 5Write the set of real numbers x with x ≤ −1 as an interval.
- 6Name the integers in the interval (−2, 3].
- 7How many natural numbers lie in [4, 10)?
- 8Is [2, 5] a subset of (1, 6)?
Show answers (try them yourself first)
- 1(1, 8)
- 2{x : −2 ≤ x < 5}
- 311 − 3 = 8
- 4No — the left end is not included.
- 5(−∞, −1]
- 6{-1, 0, 1, 2, 3}
- 76 (they are 4, 5, 6, 7, 8, 9)
- 8Yes — every point from 2 to 5 lies strictly between 1 and 6.
Universal set and Venn diagrams
When we talk about students of Class 11, the whole school is the background. A background set helps us draw a picture.
The universal set U is the big set that contains all elements under discussion. A Venn diagram shows sets as circles drawn inside a rectangle for U; elements are dots in the circles.
In the figure: The rectangle is U. The circles A and B overlap where they share elements.
Rule 1: Choose a universal set — Pick the smallest sensible set containing everything needed.
- Easy: For vowels and consonants of English: U = the alphabet.
- Medium: For even numbers and multiples of 3 up to 20: U = {1, 2, …, 20}.
- Exam-type: Different problems may use different universal sets.
Rule 2: Read a Venn diagram — Elements in the overlap belong to both sets.
- Easy: U = {1…8}, A = {1, 2, 3}, B = {3, 4, 5}: 3 sits in the overlap.
- Medium: Elements 6, 7, 8 are outside both circles.
- Exam-type: Disjoint sets have circles that do not touch.
A subset circle is drawn fully inside the bigger circle.
Forgetting elements that belong to U but to neither A nor B.
Venn = picture of sets inside the frame U.
Practice questions
- 1What does the rectangle in a Venn diagram stand for?
- 2Elements in the overlap of A and B belong to which sets?
- 3How will you draw A ⊆ B?
- 4U = {1, …, 10}, A = {2, 4, 6}. Which elements are outside A?
- 5A = {1, 2, 3} and B = {3, 4}. Which element is in the overlap?
- 6How will you draw two sets with no common element?
- 7U = {1, …, 12}, A = multiples of 3, B = multiples of 4. Elements in the overlap?
- 8For the same U, A and B, which elements lie outside both circles?
Show answers (try them yourself first)
- 1The universal set U.
- 2Both A and B.
- 3Draw circle A completely inside circle B.
- 4{1, 3, 5, 7, 8, 9, 10}
- 53
- 6As two separate circles that do not overlap.
- 7{12}
- 8{1, 2, 5, 7, 10, 11}