hsanotes.com notes – short, clear, exam-focused – help score 90+ marks in Punjab board exams.
Features of hsanotes.com notes
- Short definitions
- Step-by-step solutions
- Tricks & aids
- PCTB / PECTAA alignment
- Set notations simplified
- Past paper questions
- Revision format
- Trusted by Punjab students
For Punjab boards (Lahore, Gujranwala, Rawalpindi, Faisalabad, Bahawalpur, Multan, Sargodha, Sahiwal, DG Khan) and top schools: The City School, Beaconhouse, Roots International, Lahore Grammar School (LGS), Karachi Grammar School, Froebel’s, The Educators, Army Public School (APS), Divisional Public School, Government Model School.
Set Builder Notation (Exercise 3.1 Q1)
(i) {1,4,9,…,484} → {x | x=n², n∈N, 1≤n≤22}
(ii) {2,4,8,…,256} → {x | x=2^n, n∈N, 1≤n≤8}
(iii) {0,±1,…,±1000} → {x | x∈Z, -1000≤x≤1000}
(iv) {6,12,18,…,120} → {x | x=6n, n∈N, 1≤n≤20}
(v) {100,102,104,…,400} → {x | x even, 100≤x≤400}
(vi) {1,3,9,27,81,…} → {x | x=3^n, n∈W}
(vii) {1,2,4,5,10,20,25,50,100} → {x | x divides 100, x∈N}
(viii) {5,10,15,…,100} → {x | x=5n, n∈N, 1≤n≤20}
(ix) Integers between 100 and 1000 → {x | x∈Z, 100<x<1000}
Tabular Form (Exercise 3.1 Q2)
(i) Multiples of 3 ≤36 → {3,6,9,…,36}
(ii) 2x+1=0 → {-1/2}
(iii) Primes <12 → {2,3,5,7,11}
(iv) Divisors of 128 → {1,2,4,8,16,32,64,128}
(v) x=2n, n<8 → {2,4,8,16,32,64,128}
(vi) x+4=0, x∈N → ∅
(vii) x=x, x∈N → N
(viii) 3x+1=0, x∈Z → ∅
Proper Subsets (Exercise 3.1 Q3)
(i) {a,b,c} → {a,b}, {a}
(ii) {0,1} → {0}, {1}
(iii) N → {1,2,…}, W
(iv) Z → N, negatives
(v) Q → Z, {0}
(vi) R → Q, Z
(vii) Q between 0 and 2 → {0<x≤1 in Q}, {1,1/2,1/3}
Set with No Proper Subset (Q4)
Empty set ∅
Difference {a,b} vs {{a,b}} (Q5)
{a,b}: elements a,b
{{a,b}}: element {a,b}
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Keywords
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