Lesson Overview
Subject
Mathematics
Chapter
1 — Relations
Reference
NCERT
Dates
16–19 Mar 2026
How to Use Each Question
📝 Tab 1 — Workspace
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🎬 Tab 3 — Solution Clip
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QR Code Slots
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⚠ Common Misconceptions
✗Confusing symmetric with antisymmetric
✗Empty relation is vacuously S & T — but not Reflexive unless A = ∅
✗Transitivity — only pairs already in R need checking
✗Equivalence class [a] = set of elements, not ordered pairs
DAY 1 · MON 23 FEB 2026
Day 1 — Types of Relations
Concept Board — Day 1
Click to open · Draw diagrams & explain concepts
▶
Types of Relations
BinaryAny subset R ⊆ A × A
IdentityIₐ = { (a,a) : a ∈ A }
EmptyR = ∅ — vacuously Symmetric & Transitive
UniversalR = A×A — RST all hold
CountTotal relations from A to B: 2^(n(A)·n(B))
Q1
Write the Cartesian product A×B and find n(A×B), given A={1,2,3} and B={4,5}.
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1
A×B = {(1,4),(1,5),(2,4),(2,5),(3,4),(3,5)}
2
n(A)=3, n(B)=2 → n(A×B) = 3×2 = 6
3
∴ A×B has 6 pairs. Total possible relations from A to B = 2⁶ = 64.
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Q2
Write the Identity relation on A={p,q,r}. Is this relation reflexive? Justify.
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1
Identity Relation: Iₐ = {(p,p),(q,q),(r,r)}
2
Check Reflexive: (p,p),(q,q),(r,r) all ∈ Iₐ — every element related to itself ✓
3
∴ Yes, Identity relation is Reflexive.
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Q3
Let A={2,3,5,7}. Is R={(a,b): a²=b, a,b ∈ A} an empty relation? Justify.
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1
Need (a,b) with a²=b, both in A={2,3,5,7}
2
2²=4∉A, 3²=9∉A, 5²=25∉A, 7²=49∉A
3
No pair (a,b) satisfies a²=b with both in A
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∴ R=∅. Yes, R is an Empty Relation on A.
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Q4
Ex 1(A) Q1(a): For the relation 'is greater than', state which properties apply from: S (Symmetric), T (Transitive), R (Reflexive), E (Equivalence), N (None).
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Q5
For the relation 'is similar to' on all triangles, state all applicable properties from {S,T,R,E,N}.
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Reflexive: Every △ is similar to itself ✓
2
Symmetric: If △A~△B then △B~△A ✓
3
Transitive: If △A~△B and △B~△C then △A~△C ✓
4
∴ 'is similar to' is R, S, T → it is an Equivalence Relation. Answer: R, S, T, E
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Day 1 — Homework Questions
HW 1
Write the Universal relation on A = {1, 3, 5}. Verify that it satisfies a − b < 5 for all pairs. How many ordered pairs does it contain?
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HW 2
How many relations are possible from A = {1, 2} to B = {a, b, c}? Use the formula 2^(mn), where |A| = m, |B| = n.
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Day 1 — Homework
HW 1
Write Universal relation on A={1,3,5}; verify using a−b<5.HW 2
How many relations possible from A={1,2} to B={a,b,c}?📎 Assignments — Day 1 (Optional · Viewable by students)
A1
A2
A3
Day 1 — Video Resources
Exit Ticket
1Write one example each: empty relation and universal relation on A={1,2,3}
2Is Identity relation on A={a,b} an Equivalence Relation?
ISC Board Tags — Day 1
DAY 2 · TUE 24 FEB 2026
Day 2 — Reflexive, Symmetric & Transitive
Concept Board — Day 2
Click to open · Draw diagrams & explain concepts
▶
RST Definitions
Reflexive(a,a) ∈ R ∀ a ∈ A | Count: 2^(n²−n)
Symmetric(a,b)∈R ⟹ (b,a)∈R ⟺ R=R⁻¹ | Count: 2^((n²+n)/2)
Transitive(a,b)∈R and (b,c)∈R ⟹ (a,c)∈R
Ex 3
R = {(a, b) : b = a + 1} on {1, 2, 3, 4, 5, 6}. Check R for Reflexive, Symmetric, and Transitive properties.
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1
Write R: y=3x → R={(1,3),(2,6),(3,9),(4,12)}
2
Reflexive? Need 3a−a=0 → 2a=0 → a=0 ∉ A ✗
3
Symmetric? (1,3)∈R but 3(3)−1=8≠0, so (3,1)∉R ✗
4
Transitive? (1,3)∈R,(3,9)∈R but 3(1)−9=−6≠0 so (1,9)∉R ✗
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∴ R is neither Reflexive, nor Symmetric, nor Transitive.
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Ex 4
R = {(x, y) : x and y work at the same place} on a set of persons. Check R for Reflexive, Symmetric, and Transitive.
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1
Reflexive: a|a for every a → (a,a)∈R ✓
2
Symmetric? (1,2)∈R but is 1 divisible by 2? No ✗
3
Transitive: x|y and y|z ⟹ x|z (standard divisibility) ✓
4
∴ R is Reflexive and Transitive but NOT Symmetric.
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Ex1A · Q6
R={(a,b): a ≤ b} on ℝ. Show R is Reflexive and Transitive but NOT Symmetric.
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Reflexive: a≤a is always true ∴ (a,a)∈R ∀a∈ℝ ✓
2
Symmetric? Let a=1,b=2: (1,2)∈R but 2≤1 is false → (2,1)∉R ✗
3
Transitive: (a,b)∈R and (b,c)∈R → a≤b and b≤c → a≤c → (a,c)∈R ✓
4
∴ R is Reflexive and Transitive but NOT Symmetric.
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Day 2 — RST Examples (Set 1)
Ex 1
R = {(x, y) : 3x − y = 0} on A = {1, 2, …, 14}. Check whether R is Reflexive, Symmetric, and Transitive.
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Ex 2
R = {(x, y) : y is divisible by x} on A = {1, 2, 3, 4, 5, 6}. Determine which of the properties R / S / T hold.
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Day 2 — RST Examples (Set 3)
Ex 5
R = {(x, y) : x is exactly 7 cm taller than y} on a set of persons. Determine whether R is Reflexive, Symmetric, or Transitive.
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Ex 6
For each, check R / S / T: (a) R = {(x, y) : x is wife of y} (b) R = {(x, y) : x is father of y}.
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Day 2 — Construct Relations with Specific Properties
Ex 7
Construct a relation on a suitable set that is Symmetric but NEITHER Reflexive NOR Transitive. Verify each property explicitly.
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Ex 8
Construct a relation on a suitable set that is Transitive but NEITHER Reflexive NOR Symmetric. Verify each property explicitly.
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Ex 9
Construct a relation on {1, 2, 3} that is Reflexive and Symmetric but NOT Transitive. Display as a set of ordered pairs.
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Ex 10
Construct a relation on a suitable set that is Reflexive and Transitive but NOT Symmetric. Verify each property.
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Ex 11
Construct a relation on {1, 2} that is Symmetric and Transitive but NOT Reflexive. Verify each property.
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Day 2 — Exercise 1(A)
Q 4
Ex 1(A) Q4: Show that R = {(1, 2), (2, 1)} on {1, 2, 3} is Symmetric but neither Reflexive nor Transitive.
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Q 5
R on {1,2,3,4} = {(1,2),(2,2),(1,1),(4,4),(1,3),(3,3),(3,2)}. Choose the correct property from options (a)–(d).
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Q 9
Is the relation 'is the square of' defined on ℕ an Equivalence Relation? Justify with proof or counterexample.
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Day 2 — Homework Questions
HW Q1
Ex 1(A): (a) For 'is the cube of' on ℝ — state all applicable properties. (b) For 'is the sister of' on a set of children — state all applicable properties.
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HW Q2
Write a relation which is: (a) only Transitive (b) only Symmetric (c) only Reflexive and Transitive (d) only Symmetric and Reflexive.
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HW Q7
R = {(a, b) : a ≤ b²} on ℝ. Show that R is neither Reflexive, nor Symmetric, nor Transitive.
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Day 2 Homework
HW 1
R={(a,b): a≤b²} on ℝ. Show R is neither R, S, nor T.HW 2
Write a relation which is: (a) only T (b) only S (c) only R and T (d) only S and R.📎 Assignments — Day 2 (Optional · Viewable by students)
A1
A2
A3
Day 2 — Video Resources
Exit Ticket
1Symmetric but neither R nor T — give one example.
2R={(1,2),(2,1)} on {1,2,3}: Is R symmetric? Reflexive? Transitive?
Mark Scheme Note
State property → assume (a,b)∈R → show consequence → conclude. Every step earns marks.
DAY 3 · WED 25 FEB 2026
Day 3 — Equivalence Relations & Classes
Concept Board — Day 3
Click to open · Draw diagrams & explain concepts
▶
Equivalence Relation
DefinitionR is ER ⟺ Reflexive AND Symmetric AND Transitive
Class [a]{b ∈ A : (b,a) ∈ R} — set of all elements related to a
PartitionClasses are mutually disjoint; their union = A
ISC NoteAlways write conclusion sentence — 1 mark awarded by ISC for it
Ex 7
R={(a,b): (a−b) divisible by 5} on ℤ. Prove R is ER. Write equivalence classes [0],[1],[2].
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Reflexive: a−a=0=5×0, so 5|(a−a) ∴ (a,a)∈R ✓
2
Symmetric: 5|(a−b) ⟹ 5|−(a−b)=(b−a) ∴ (b,a)∈R ✓
3
Transitive: 5|(a−b) and 5|(b−c) ⟹ 5|(a−b+b−c)=5|(a−c) ✓
4
Conclusion: Since R is Reflexive, Symmetric and Transitive → R is an Equivalence Relation.
5
[0]={…,−10,−5,0,5,10,…} [1]={…,−4,1,6,11,…} [2]={…,−3,2,7,12,…}
6
There are 5 distinct classes [0],[1],[2],[3],[4] — they partition ℤ.
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CBSE 2024
R on A={−4,…,4}, R={(x,y): x+y divisible by 2}. Show R is ER; write equivalence class [2].
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Reflexive: x+x=2x, divisible by 2 ✓
2
Symmetric: 2|(x+y) ⟹ 2|(y+x) ✓
3
Transitive: 2|(x+y) and 2|(y+z) ⟹ 2|(x+z+2y) ⟹ 2|(x+z) ✓
4
∴ R is an Equivalence Relation.
5
[2] = all x where x+2 divisible by 2, i.e. x even: [2]={−4,−2,0,2,4}
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Day 3 — Equivalence Relations (Examples Set 1)
Ex 4
R = {(x, y) : x and y have the same number of pages} on all books in a library. Show that R is an Equivalence Relation.
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Ex 5
The congruence relation '≅' on the set of all triangles in Euclidean geometry. Show that it is an Equivalence Relation.
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Day 3 — Equivalence Relations (Examples Set 2)
Ex 6
The subset relation '⊂' with respect to sets. Show that '⊂' is NOT an Equivalence Relation. Identify which property fails.
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Ex 8
S = {(a, b) : |a − b| is divisible by 4} on A = {x ∈ ℤ : 0 ≤ x ≤ 12}. Prove S is an ER. Find all elements related to 1.
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Ex 9
R = {(P, Q) : OP = OQ} in a plane (O = origin). Show R is an ER. Describe the set of all points related to P ≠ (0, 0).
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Day 3 — Equivalence Relations (Examples Set 3 & 4)
Ex 10
Show that the relation '≥' on ℝ is NOT an Equivalence Relation. Identify which property fails and give a counterexample.
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Ex 12
(i) R = {(T₁, T₂) : T₁ is similar to T₂} — show R is an ER. (ii) Among T₁(3,4,5), T₂(5,12,13), T₃(6,8,10) — which triangles are related?
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Day 3 — Exercise 1(A)
Q 3
Prove that the relation 'is brother of' on the set of all family members is a Transitive relation.
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Q 11
R = {(P₁, P₂) : P₁ and P₂ have the same number of sides} on all polygons. Show R is an ER. Find the equivalence class of right-triangle T(3, 4, 5).
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Q 12
R = {(L₁, L₂) : L₁ ∥ L₂} on all lines in the XY-plane. Show R is an ER. Find the set of all lines related to y = 2x + 4.
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Q 15
R = {(a, b) : |a − b| is even} on A = {1, 2, 3, 4, 5}. Prove that R is an Equivalence Relation.
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Q 16
aRb iff (a + b) is even, a, b ∈ ℤ. Prove that R is an Equivalence Relation.
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Day 3 — Homework Questions
HW Ex11
For each relation, write YES or NO for being an Equivalence Relation: (a) parallel to (b) perpendicular to (c) greater than (d) factor of (e) multiple of.
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Day 3 Homework
HW 1
Prove R={(L₁,L₂): L₁∥L₂} on all lines in XY-plane is an ER. Find all lines related to y=2x+4.HW 2
aRb iff (a+b) is even, a,b∈ℤ. Prove R is an ER.📎 Assignments — Day 3 (Optional · Viewable by students)
A1
A2
A3
Day 3 — Video Resources
Exit Ticket
1What is [3] under R={(a,b): 3|(a−b)} on ℤ?
2Can two equivalence classes overlap? Prove or disprove.
6M Mark Scheme
1M Reflexive + 1M Symmetric + 1M Transitive + 1M Conclusion + 2M for Equivalence Class
DAY 4 · THU 26 FEB 2026
Day 4 — ℕ×ℕ Equivalence & Review
Concept Board — Day 4
Click to open · Draw diagrams & explain concepts
▶
Equivalence on ℕ×ℕ
Standard(a,b)R(c,d) ⟺ ad=bc (cross-multiplication form of a/b=c/d)
ISC 2024(a,b)R(c,d) iff a−c=b−d → same as a−b=c−d
Class[(2,6)]={(1,3),(2,6),(3,9),…} all pairs in ratio 1:3
Ex 13
R on ℕ×ℕ: (a,b)R(c,d) ⟺ ad=bc. Prove R is ER. Find [(2,6)].
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Reflexive: (a,b)R(a,b)? Need ab=ba ✓ (commutative)
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Symmetric: ad=bc ⟹ cb=da ∴ (c,d)R(a,b) ✓
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Transitive: (a,b)R(c,d): ad=bc. (c,d)R(e,f): cf=de.
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Multiply: (ad)(cf)=(bc)(de) ⟹ acdf=bcde ⟹ af=be (divide by cd) ∴ (a,b)R(e,f) ✓
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∴ R is an Equivalence Relation on ℕ×ℕ.
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[(2,6)]: need 2b=6a → a/b=1/3 ∴ [(2,6)]={(1,3),(2,6),(3,9),(4,12),…}
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Ex 19
R on A×A (A={1..9}): (a,b)R(c,d) iff a+d=b+c. Show R is ER. Find [(2,5)].
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Note: a+d=b+c ⟺ a−b=c−d (same difference)
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Reflexive: a+b=b+a ✓
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Symmetric: a+d=b+c ⟹ c+b=d+a ✓
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Transitive: a+d=b+c and c+f=d+e → adding → a+f=b+e ✓
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∴ R is an Equivalence Relation. Class depends on a−b=2−5=−3.
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[(2,5)]={(1,4),(2,5),(3,6),(4,7),(5,8),(6,9)}
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Day 4 — Equivalence on ℕ×ℕ (Examples Set 1 & 2)
Ex 14
R = {(a, b) : a − b < 5} on ℝ. Is R an Equivalence Relation? Justify by checking all three properties.
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Ex 15
R = {(a, b) : a − b is even, a, b ∈ ℤ}. Show that R is an Equivalence Relation.
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Ex 17
(a, b) R (c, d) iff ad(b + c) = bc(a + d) on ℕ×ℕ. Show that R is an Equivalence Relation.
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Ex 18
R = {(a, b) : a = b, 0 ≤ a ≤ 12} on A = {0, 1, …, 12}. Show R is an ER. Find all elements related to 1.
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Day 4 — CBSE 2024 & Exercise 1(A)
CBSE Q2
CBSE 2024: (a, b) R (c, d) iff a − c = b − d on ℕ×ℕ. Show that R is an Equivalence Relation.
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Q 10
Ex 1(A) Q10(a)–(e): State S / T / R / E / N for each: (a) smaller than (b) father of (c) parallel to (d) multiple of (e) congruent to.
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Q 14
Ex 1(A) Q14: R = {(x, y) : x + y = 10, x, y ∈ ℕ}. Is R Reflexive, Symmetric, Transitive? Examine each property separately.
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Day 4 — Chapter Review: Very Short Answer (VSA)
VSA Q1
Fill in the blank: 'A relation R in set A is called _______ if (a₁, a₂) ∈ R ⟹ (a₂, a₁) ∈ R.'
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VSA Q6
Write the smallest Reflexive relation on A = {a, b, c}. How many ordered pairs does it contain?
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VSA Q8
How many Reflexive relations are possible in a set A with n(A) = 3? Use the formula 2^(n²−n).
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VSA Q9
R = {(1,1),(1,2),(2,2),(3,3)} on S = {1,2,3}. Which element(s) must be removed from R to make R an Equivalence Relation?
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VSA Q11
R = {(x, y) : x + 2y = 8} on ℕ. Write all ordered pairs in R and find the Range of R.
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VSA Q12
R = {(a, a³) : a is a prime number less than 5}. List all ordered pairs in R and find the Range of R.
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Day 4 — True or False
T/F Q16
True or False: R = {(3,1),(1,3),(3,3)} on A = {1,2,3} is Symmetric and Transitive but NOT Reflexive. Justify your answer.
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T/F Q17
True or False: R = {(1,1),(1,2),(2,1),(3,3)} on {1,2,3} is an Equivalence Relation. Identify any missing ordered pairs.
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T/F Q19
True or False: Every relation that is Symmetric and Transitive is also Reflexive. Prove or give a counterexample.
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Day 4 — Homework & Self-Study
HW 1
NCERT Exemplar: For all a, b ∈ ℤ, define aRb iff (a − b) is divisible by n. Show that R is an Equivalence Relation. Also complete all pending Ex 1(A) questions.
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HW 2
Attempt Assertion–Reason questions Q21(i) to Q21(vi) from the textbook.
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HW 3
Read Case Study Q24 (General Elections context) from the textbook and answer ALL parts of the case study carefully.
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Day 4 Homework
HW 1
Complete all pending Ex1(A). NCERT Exemplar: aRb iff (a−b) divisible by n on ℤ — show R is ER.HW 2
Attempt Assertion–Reason Q21(i)–(vi) from textbook.📎 Assignments — Day 4 (Optional · Viewable by students)
A1
A2
A3
HW 3
Read Case Study Q24 (General Elections context) and answer all parts.Day 4 — Video Resources
Final Exit Ticket
1In one sentence: what makes a relation an equivalence relation?
2Write [(1,2)] under (a,b)R(c,d) iff a+d=b+c on ℕ×ℕ.
3How many equivalence relations on {1,2,3}? (Bell number)
Chapter Summary
R(a,a)∈R ∀a
S(a,b)∈R ⟹ (b,a)∈R
T(a,b),(b,c)∈R ⟹ (a,c)∈R
ER+S+T simultaneously
[a]Disjoint partition of A
CRITICAL THINKING · Chapter 1 — Relations
Critical Thinking — ISC Level Questions
Concept Board — Critical Thinking
Click to open · Notes, diagrams, working space
▶
Section A — Multiple Choice Questions (1 mark each)
MCQ Q1 · 1M
Which of the following relations on ℤ is an equivalence relation? (a) R = {(a,b) : a > b} (b) R = {(a,b) : |a − b| ≤ 1} (c) R = {(a,b) : a − b is divisible by 5} (d) R = {(a,b) : a ≠ b}
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MCQ Q2 · 1M
The relation R = {(1,1), (2,2), (3,3)} on A = {1, 2, 3} is: (a) symmetric only (b) transitive only (c) reflexive only (d) an equivalence relation
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MCQ Q3 · 1M
Let R be a relation on ℝ defined by (a, b) ∈ R ⟺ a ≤ b. Which property does R LACK? (a) reflexivity (b) transitivity (c) anti-symmetry (d) symmetry
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MCQ Q4 · 1M
The number of equivalence relations that can be defined on the set {1, 2, 3} is: (a) 2 (b) 3 (c) 5 (d) 6
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MCQ Q5 · 1M
If R and S are both equivalence relations on set A, then which of the following is also an equivalence relation on A? (a) R ∪ S (b) R − S (c) R ∩ S (d) R × S
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MCQ Q6 · 1M
The empty relation ∅ on a non-empty set A is: (a) reflexive and symmetric (b) symmetric and transitive (c) reflexive only (d) an equivalence relation
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MCQ Q7 · 1M
R is a relation on A = {1, 2, 3, 4} defined by R = {(1,2), (2,3), (1,3)}. R is: (a) reflexive and transitive (b) transitive only (c) symmetric only (d) neither reflexive nor symmetric
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MCQ Q8 · 1M
If n(A) = 3, the maximum number of elements a reflexive relation on A can have is: (a) 3 (b) 6 (c) 9 (d) 12
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Section B — Assertion & Reason (1 mark each)
AR Q9 · 1M
Assertion (A): The identity relation on A = {1, 2, 3} is an equivalence relation. | Reason (R): A relation is an equivalence relation if it is reflexive, symmetric, and transitive. | (a) Both A and R are true and R is the correct explanation. (b) Both A and R are true but R is NOT the correct explanation. (c) A is true but R is false. (d) A is false but R is true.
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AR Q10 · 1M
Assertion (A): Every reflexive relation on a set A is also symmetric. | Reason (R): Reflexivity only guarantees (a,a) ∈ R for all a, but says nothing about (a,b) and (b,a) for a ≠ b. | (a) Both A and R are true and R is the correct explanation. (b) Both A and R are true but R is NOT the correct explanation. (c) A is true but R is false. (d) A is false but R is true.
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AR Q11 · 1M
Assertion (A): The universal relation U = A × A on any non-empty set A is an equivalence relation. | Reason (R): In U, every pair (a,b) is in R, so (a,a), (a,b) ⇒ (b,a), and (a,b) and (b,c) ⇒ (a,c) all hold. | (a) Both A and R are true and R is the correct explanation. (b) Both A and R are true but R is NOT the correct explanation. (c) A is true but R is false. (d) A is false but R is true.
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AR Q12 · 1M
Assertion (A): The relation R = {(a,b) : a and b are born in the same city} on the set of all people is an equivalence relation. | Reason (R): The relation partitions the set of people into disjoint equivalence classes, one for each city. | (a) Both A and R are true and R is the correct explanation. (b) Both A and R are true but R is NOT the correct explanation. (c) A is true but R is false. (d) A is false but R is true.
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Section C — Short Answer (2 marks each)
SA Q13 · 2M
Let A = {1, 2, 3}. Write down a relation on A that is: (a) Reflexive and symmetric but NOT transitive. (b) Transitive but neither reflexive nor symmetric. Justify each with a brief reason.
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SA Q14 · 2M
Define the relation R on ℤ by: (a,b) ∈ R ⟺ 2a + 3b is divisible by 5. Check whether R is (i) reflexive (ii) symmetric (iii) transitive.
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SA Q15 · 2M
Let R be an equivalence relation on A = {1,2,3,4,5,6} defined by R = {(a,b) : 3 divides (a−b)}. Find all equivalence classes of R. How many are there, and do they partition A?
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SA Q16 · 2M
A student defines a relation R on ℝ by R = {(a,b) : a² = b²}. Show R is an equivalence relation. Describe the equivalence class of 3 and the class of 0.
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SA Q17 · 2M
Let R and S be two equivalence relations on a set A. Prove that R ∩ S is also an equivalence relation on A. Give a counterexample to show R ∪ S need NOT be an equivalence relation.
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SA Q18 · 2M
On the set of all lines in a plane, define: R₁ = {(l₁,l₂) : l₁ is parallel to l₂} and R₂ = {(l₁,l₂) : l₁ is perpendicular to l₂}. Which of R₁ or R₂ is an equivalence relation? Justify. Describe the equivalence class of y = 2x + 1 under R₁.
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Section D — Long Answer (4 marks each)
LA Q19 · 4M
Let A = ℤ. Define R on A by (a,b) ∈ R ⟺ a − b is divisible by n (for a fixed positive integer n). (i) [1M] Prove R is reflexive. (ii) [1M] Prove R is symmetric. (iii) [1M] Prove R is transitive. (iv) [1M] For n = 3, find all equivalence classes. How many distinct classes exist?
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LA Q20 · 4M
Let A = {1,2,3,4,5,6,7,8}. Define R = {(a,b) : |a−b| is even}. (i) [1M] Prove R is an equivalence relation. (ii) [1M] Find all equivalence classes. (iii) [1M] Show the equivalence classes partition A. (iv) [1M] A student claims [1] = [3] as equivalence classes. Is this correct? Justify.
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LA Q21 · 4M
On ℝ × ℝ, define R by (a,b) R (c,d) ⟺ a + d = b + c. (i) [2M] Prove R is an equivalence relation. (ii) [1M] Find the equivalence class of (1,2). (iii) [1M] Show the equivalence classes correspond to lines of the form y − x = k (k ∈ ℝ). Interpret geometrically.
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Section E — Case Study (1 mark each)
CS Q22 · 1M
CASE STUDY — School Timetable: A school has 120 students in Class XII. Define R on all students by: (x,y) ∈ R ⟺ x and y study the same elective combination. Electives: PCM (40 students), PCB (35), Commerce-Maths (30), Humanities (15). | | (i) Show that R is an equivalence relation.
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CS Q23 · 1M
CASE STUDY (contd.) — (ii) Identify the equivalence classes. How many are there? What does each class represent?
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CS Q24 · 1M
CASE STUDY (contd.) — (iii) Verify that the equivalence classes partition the set of 120 students. Show the partition adds up correctly.
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CS Q25 · 1M
CASE STUDY (contd.) — (iv) A new student joins and studies PCM. Without checking every other student, how can you determine their equivalence class? What property of equivalence classes guarantees this?
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Section F — HOTS & Proof (5 marks each)
HOTS Q26 · 5M
A relation R on set A is called "anti-symmetric" if (a,b) ∈ R and (b,a) ∈ R ⇒ a = b. (i) [1M] Is the identity relation on ℤ anti-symmetric? Justify. (ii) [1M] Can a relation be BOTH symmetric AND anti-symmetric? Construct an example or prove impossibility. (iii) [2M] The relation "≤" on ℝ is anti-symmetric. Prove this. Is it also an equivalence relation? What property does it lack? (iv) [1M] A student claims: "If R is reflexive and anti-symmetric, then R is a partial order." State the one additional property needed and give an example.
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HOTS Q27 · 5M
Let A be any non-empty set and 𝒫(A) be its power set. Define X R Y ⟺ X ⊆ Y. (i) [1M] Show R is reflexive and transitive but NOT symmetric (for |A| ≥ 2). (ii) [1M] Is R anti-symmetric? Prove it. (iii) [2M] For A = {1,2}, write out 𝒫(A). List ALL ordered pairs (X,Y) in R. Draw the full arrow diagram. (iv) [1M] Does R form a partition on 𝒫(A)? Why or why not? What concept replaces 'partition' here?
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Section G — Diagram-Based (2–3 marks each)
DQ Q28 · 3M
Four relations on A = {1,2,3}: R₁={(1,1),(2,2),(3,3),(1,2),(2,1),(2,3),(3,2),(1,3),(3,1)}, R₂={(1,1),(2,2),(3,3),(1,2),(2,3)}, R₃={(1,2),(2,1)}, R₄=∅. For EACH relation check Reflexive, Symmetric, Transitive and state the type (equivalence / partial order / none etc.).
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DQ Q29 · 3M
Arrow diagram of R on A={1,2,3,4}: pairs (1,1),(2,2),(3,3),(4,4),(1,2),(2,1),(3,4),(4,3). (i)[1M] Is R reflexive? Verify for each element. (ii)[1M] Is R symmetric? Identify a pair (a,b)∈R and verify (b,a)∈R. (iii)[1M] Is R transitive? Check (1,2),(2,1)→(1,1) and (3,4),(4,3). (iv) Is R an equivalence relation? Find equivalence classes and verify they partition A.
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DQ Q30 · 3M
Equivalence relation R on A={a,b,c,d,e,f}: self-loops for all + {a,b,c} fully connected + {d,e,f} fully connected. (i)[1M] Identify equivalence classes directly from the diagram. (ii)[1M] What is [a]? What is [d]? Show [a] ∩ [d] = ∅ and [a] ∪ [d] = A. (iii)[1M] How many ordered pairs does R contain in total? Count using equivalence classes.
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DQ Q31 · 3M
INCOMPLETE relation R on A={1,2,3,4}: current pairs (1,1),(2,2),(1,2),(2,1),(2,3). Some pairs are missing. (i)[1M] Identify ALL missing pairs needed to make R reflexive. (ii)[1M] After fixing reflexivity, identify ALL missing pairs for symmetry. (iii)[1M] After fixing symmetry, identify ALL pairs for transitivity. List the final complete R. (iv) What are the equivalence classes?
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DQ Q32 · 3M
Two students define: R₁={(a,b) : a and b are both odd or both even} and R₂={(a,b) : a ≡ b (mod 2)} on A={1,2,3,4,5,6}. (i)[1M] Are R₁ and R₂ the same relation? Justify by comparing ordered pair sets. (ii)[1M] Find the equivalence classes of R₁ (= R₂). How many classes? (iii)[1M] How many ordered pairs does this equivalence relation contain? Use: sum of |class|².
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Critical Thinking — Video Resources