Class 12 Relations and Functions Exercise 1.1 Solutions

Class 12 Relations and Functions Exercise 1.1 Solutions
Class 12 — Chapter 1

Relations & Functions — Exercise 1.1 (Solutions)

Concise answers for reflexive, symmetric and transitive properties — ready for quick revision and copy-paste into notes.

Focus: Class 12 Relations & Functions Exercise 1.1

Quick Summary

This page contains concise solutions to the 14 questions from Exercise 1.1 (Reflexive, Symmetric, Transitive). Click any question to expand the answer.

Questions & Concise Solutions

Q1. On A = {1,...,14}, relation R = {(x,y) : 3x - y = 0} (i.e., y = 3x). Is R reflexive, symmetric, transitive?

Reflexive? No. For reflexivity we’d need (x,x): x = 3x ⇒ x = 0, which is not in A.

Symmetric? No. If (x,y) ∈ R then y = 3x. For (y,x) to hold we’d need x = 3y = 9x ⇒ x=0 only.

Transitive? No. Example: (1,3) ∈ R and (3,9) ∈ R but (1,9) ∉ R because 9 ≠ 3·1.

Conclusion: Not reflexive, not symmetric, not transitive.

Q2. On ℕ, R = {(x,y): y = x+5 and x < 4}. Reflexive, symmetric, transitive?

Possible pairs: (1,6), (2,7), (3,8).

Reflexive? No. No (x,x) satisfies x = x+5.

Symmetric? No (e.g., (1,6) ∈ R but (6,1) ∉ R).

Transitive? Yes (vacuously) — there is no pair whose first coordinate is 6,7,8, so antecedent of transitivity never holds.

Q3. On A = {1,..,6}, R = {(x,y): y divisible by x}. Reflexive, symmetric, transitive?

Reflexive: Yes (x|x).

Symmetric: No (2|4 but 4 ∤ 2).

Transitive: Yes (divisibility is transitive: x|y and y|z ⇒ x|z).

Conclusion: Reflexive & transitive, not symmetric.

Q4. On ℤ, R = {(x,y): x – y is an integer}. Reflexive, symmetric, transitive?

For integers x,y, x – y is always an integer ⇒ R = ℤ × ℤ (universal relation).

Universal relation is reflexive, symmetric and transitive.

Q5. For people in a town — classify these relations: (a) work at same place (b) live in same locality (c) x is exactly 7 cm taller than y (d) x is wife of y (e) x is father of y

(a) Work at same place: Equivalence (reflexive, symmetric, transitive).

(b) Live in same locality: Equivalence as well.

(c) Exactly 7 cm taller: Not reflexive, not symmetric, not transitive.

(d) x is wife of y: Generally not reflexive, not symmetric (wife ≠ wife-of symmetric unless wording includes spouse), not transitive.

(e) x is father of y: Not reflexive, not symmetric, not transitive.

Q6. On ℝ, R = {(a,b): a ≤ b²}. Reflexive, symmetric, transitive?

Reflexive? No (take a = 1/2: 1/2 ≤ (1/2)² is false).

Symmetric? No (0 ≤ 1² but 1 ≤ 0² false).

Transitive? No (counterexamples exist — chaining may fail).

Q7. On {1..6}, R = {(a,b): b = a+1}. Reflexive, symmetric, transitive?

Pairs: (1,2),(2,3),(3,4),(4,5),(5,6).

Not reflexive, not symmetric, not transitive (since (1,2) & (2,3) ∈ R but (1,3) ∉ R).

Q8. On ℝ, R = {(a,b): a ≤ b}. Reflexive, symmetric, transitive?

Reflexive: yes. Transitive: yes. Symmetric: no (1 ≤ 2 but 2 ≤ 1 false).

Q9. On ℝ, R = {(a,b): a ≤ b³}. Reflexive, symmetric, transitive?

Not reflexive (small positive a can fail), not symmetric, not transitive in general.

Q10. On {1,2,3}, R = {(1,2),(2,1)}. Reflexive, symmetric, transitive?

Symmetric: yes. Reflexive: no (missing diagonal entries). Transitive: no ( (1,2) & (2,1) ∈ R but (1,1) ∉ R ).

Q11. Books in a library: R = {(x,y): x and y have same number of pages}. Show it’s an equivalence relation.

Same number of pages is reflexive, symmetric and transitive (equality of page-count). Hence an equivalence relation. Equivalence classes = books grouped by page count.

Q12. On {1,2,3,4,5}, R = {(a,b): |a-b| is even}. Show equivalence and classes.

Reflexive: yes. Symmetric: yes. Transitive: yes (parity). Classes: {1,3,5} and {2,4}.

Q13. On {0..12}: (i) |a-b| multiple of 4 — class of 1? (ii) a = b — class of 1?

(i) Congruence mod 4: classes: {0,4,8,12}, {1,5,9}, {2,6,10}, {3,7,11}. Class of 1 = {1,5,9}.

(ii) Equality relation: class of 1 = {1}.

Q14. Examples of relations with combos of properties (i)-(v).

(i) Symmetric but neither reflexive nor transitive: R = {(1,2),(2,1)} on {1,2}.

(ii) Transitive but neither reflexive nor symmetric: R = {(1,2),(2,3),(1,3)}.

(iii) Reflexive & symmetric but not transitive: diagonal ∪ {(1,2),(2,1),(2,3),(3,2)} but omit (1,3).

(iv) Reflexive & transitive but not symmetric: partial order like {(1,1),(2,2),(1,2)}.

(v) Symmetric & transitive but not reflexive: empty relation ∅ (vacuously symmetric & transitive but not reflexive).

Made with ❤️ for quick revision — MathScience.in

Unit I: Diversity of Living Organisms

  1. The Living World
  2. Biological Classification
  3. Plant Kingdom
  4. Animal Kingdom

Unit II: Structural Organisation in Animals and Plants
5. Morphology of Flowering Plants
6. Anatomy of Flowering Plants
7. Structural Organisation in Animals

Unit III: Cell: Structure and Function
8. Cell – The Unit of Life
9. Biomolecules
10. Cell Cycle and Cell Division

Unit IV: Plant Physiology
11. Transport in Plants
12. Mineral Nutrition
13. Photosynthesis in Higher Plants
14. Respiration in Plants
15. Plant Growth and Development

Unit V: Human Physiology
16. Digestion and Absorption
17. Breathing and Exchange of Gases
18. Body Fluids and Circulation
19. Excretory Products and their Elimination
20. Locomotion and Movement
21. Neural Control and Coordination
22. Chemical Coordination and Integration

Class 11 Biology NCERT Solution – New NCERT Syllabus (2024–25)

These 14 chapters are included in the official CBSE syllabus for the 2024–25 board exam.

Unit I: Diversity of Living Organisms

  1. The Living World
  2. Biological Classification
  3. Plant Kingdom
  4. Animal Kingdom

Unit II: Structural Organisation in Animals and Plants

  1. Morphology of Flowering Plants

Note: “Anatomy of Flowering Plants” and “Structural Organisation in Animals” have been removed from the CBSE syllabus.

Unit III: Cell Structure and Function

  1. Cell: The Unit of Life
  2. Biomolecules
  3. Cell Cycle and Cell Division

Unit IV: Plant Physiology

  1. Photosynthesis in Higher Plants
  2. Respiration in Plants
  3. Plant Growth and Development

Unit V: Human Physiology

  1. Breathing and Exchange of Gases
  2. Body Fluids and Circulation
  3. Excretory Products and Their Elimination

Chapters like “Locomotion and Movement”, “Neural Control and Coordination”, and “Chemical Coordination and Integration” are excluded from the new syllabus.

Total Chapters in New Syllabus: 14

These are the only chapters needed for the 2024–25 CBSE board exam.

Chapters in NCERT Textbook But Excluded from NEW CBSE Syllabus (2024–25)

These chapters are part of the NCERT book but are not included in the CBSE 2024–25 exam syllabus. Still important for NEET.

Chapters Excluded from CBSE Class 11 Biology Syllabus (2024–25)

  1. Anatomy of Flowering Plants
  2. Structural Organisation in Animals
  3. Locomotion and Movement
  4. Neural Control and Coordination
  5. Chemical Coordination and Integration

Important Notes:

  • These chapters still exist in the NCERT textbook, but are not required for the CBSE 2024–25 exam.
  • Do not prepare or publish these chapters as part of the main syllabus for CBSE board content.

Important Notes:

  • Only 14 chapters are to be studied for the Class 11 CBSE exam (2024–25).
  • The remaining 5 chapters are not for board exams, but are useful for NEET and concept clarity.
NEET BIOLOGY

Class-wise Solutions

Class 12:

Class 12 Physics – NCERT Solutions

Class 11:

Class 10:

Class 9:

Class 8:

Class 7:

Class 6:

Subject-wise Solutions

Physics:

Chemistry:

Biology:

Math:

Science:

NEET BIOLOGY

For additional reference and to access the official NCERT Class 10 Maths textbook, visit the NCERT website. This will help you understand the concepts covered in Class 12 Relations and Functions Exercise 1.1 more effectively.

🔗 Visit NCERT Official Website: NCERT Class 10 Maths Book

Class 12 Relations and Functions Exercise 1.1