Muft Shiksha
Update

Muft Shiksha™ एक 100% Free Education Portal है 🇮🇳, जिसका उद्देश्य Class 9–12 के हर विद्यार्थी तक High-Quality Education को पूरी तरह मुफ्त पहुँचाना है। 🇮🇳 हम मानते हैं कि अच्छी शिक्षा किसी student की आर्थिक स्थिति पर निर्भर नहीं होनी चाहिए। 🇮🇳 हर विद्यार्थी को वही Quality Study Material, MCQs, Quizzes, Exam Preparation, Concept-Based Learning और Bilingual Support मिलना चाहिए, जो आमतौर पर महंगी Coaching या Premium Platforms में मिलता है। Muft Shiksha™ 🇮🇳 इसी सोच के साथ बनाया गया है

Subjects
Class 12MathematicsRelations and FunctionsSymmetric relationHardLevel 12
0 reads0 ratings0 helpful

If R is symmetric, which statement about A×A−R is correct?

Advertisement

Answer and explanation

Correct answer: It will also be symmetric

Let S=(A×A)−R be the complement of R in A×A. To test symmetry, assume (a,b)∈S, so (a,b)∉R. If (b,a) were in R, then symmetry of R would imply (a,b)∈R, contradicting the assumption. Therefore (b,a)∉R and hence (b,a)∈S. Thus S is symmetric. The complement need not be empty or diagonal-only, and it can be symmetric even when R is neither universal nor empty.

Related tags

Complement RelationSymmetric RelationProof By Contradiction

Frequently asked questions

What is the correct answer to this question?

It will also be symmetric

Why is this the correct answer?

Let S=(A×A)−R be the complement of R in A×A. To test symmetry, assume (a,b)∈S, so (a,b)∉R. If (b,a) were in R, then symmetry of R would imply (a,b)∈R, contradicting the assumption. Therefore (b,a)∉R and hence (b,a)∈S. Thus S is symmetric. The complement need not be empty or diagonal-only, and it can be symmetric even when R is neither universal nor empty.

Which subject and chapter does this question cover?

This is a Class 12 Mathematics question. Chapter: Relations and Functions. Topic: Symmetric relation.

Was this question useful?

No ratings yetWrite a review / Rate this question

Student Reviews

No published reviews yet.

KEEP PRACTISING

10 more MCQs
  1. 1On the set (A={1,2,3}), the relation (R={(1,2),(2,1),(3,3)}) is given. What type of relation is it?
  2. 2If a relation (R) contains ((4,7)) and (R) is symmetric, which ordered pair must definitely be in (R)?
  3. 3On (A={1,2}), (R={(1,1),(1,2)}) is given. Which pair must be added to make it symmetric?
  4. 4Which statement correctly identifies a symmetric relation?
  5. 5On the set (A={1,2,3}), which relation is not symmetric?
  6. 6If (R={(2,2),(3,5),(5,3)}), which statement about (R) is correct?
  7. 7On (A={1,2,3}), (R={(1,2),(2,1),(2,3)}). Why is (R) not symmetric?
  8. 8Is the empty relation (R=\varnothing) on a non-empty set (A) symmetric or not?
  9. 9Is the universal relation (R=A \times A) on a set (A) symmetric or not?
  10. 10On (A={1,2,3}), (R={(1,1),(2,2),(3,3)}). Why is this relation symmetric?
Advertisement