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Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R is
- a)Non commutative relation
- b)A universal relation
- c)An equivalence relation
- d)An empty relation
Correct answer is option 'C'. Can you explain this answer?
Most Upvoted Answer
Let T be the set of all triangles in a plane with R is a relation in T...
For reflexive, T1 is congruent to T1
⇒ (T1,T1) Î R For symmetric, (T1,T2) ∈ R ⇒ T1 is congruent to T2 ⇒ T2 is congruent to T1 ⇒ (T2,T1) ∈ R. Hence it is symmetric.
For transitive, (T1, T2) ∈ R ⇒ T1 is congruent to T2 and (T2,T3) ∈ R ⇒ T2 is congruent to T3 which implies T1 is congruent to T3 ⇒ (T1,T3) ∈ R. Hence, it is transitive.
Hence, R is an equivalence relation.
⇒ (T1,T1) Î R For symmetric, (T1,T2) ∈ R ⇒ T1 is congruent to T2 ⇒ T2 is congruent to T1 ⇒ (T2,T1) ∈ R. Hence it is symmetric.
For transitive, (T1, T2) ∈ R ⇒ T1 is congruent to T2 and (T2,T3) ∈ R ⇒ T2 is congruent to T3 which implies T1 is congruent to T3 ⇒ (T1,T3) ∈ R. Hence, it is transitive.
Hence, R is an equivalence relation.
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Let T be the set of all triangles in a plane with R is a relation in T...
Equivalence Relation Explanation:
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Let's see how relation R in set T satisfies these properties:
Reflexive:
- For any triangle T1, T1 is congruent to itself (by definition of congruence).
- Therefore, R contains pairs like (T1, T1) for all T1 in T, making it reflexive.
Symmetric:
- If triangle T1 is congruent to triangle T2, then triangle T2 is also congruent to triangle T1 (by the properties of congruence).
- This means that if (T1, T2) is in R, then (T2, T1) must also be in R, satisfying symmetry.
Transitive:
- If triangle T1 is congruent to triangle T2, and triangle T2 is congruent to triangle T3, then triangle T1 is congruent to triangle T3 (by the transitive property of congruence).
- This implies that if (T1, T2) and (T2, T3) are in R, then (T1, T3) must also be in R, meeting the transitive criterion.
Since relation R in set T satisfies the properties of reflexivity, symmetry, and transitivity, it is an equivalence relation.
An equivalence relation is a relation that is reflexive, symmetric, and transitive. Let's see how relation R in set T satisfies these properties:
Reflexive:
- For any triangle T1, T1 is congruent to itself (by definition of congruence).
- Therefore, R contains pairs like (T1, T1) for all T1 in T, making it reflexive.
Symmetric:
- If triangle T1 is congruent to triangle T2, then triangle T2 is also congruent to triangle T1 (by the properties of congruence).
- This means that if (T1, T2) is in R, then (T2, T1) must also be in R, satisfying symmetry.
Transitive:
- If triangle T1 is congruent to triangle T2, and triangle T2 is congruent to triangle T3, then triangle T1 is congruent to triangle T3 (by the transitive property of congruence).
- This implies that if (T1, T2) and (T2, T3) are in R, then (T1, T3) must also be in R, meeting the transitive criterion.
Since relation R in set T satisfies the properties of reflexivity, symmetry, and transitivity, it is an equivalence relation.
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Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R isa)Non commutative relationb)A universal relationc)An equivalence relationd)An empty relationCorrect answer is option 'C'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R isa)Non commutative relationb)A universal relationc)An equivalence relationd)An empty relationCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R isa)Non commutative relationb)A universal relationc)An equivalence relationd)An empty relationCorrect answer is option 'C'. Can you explain this answer?.
Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R isa)Non commutative relationb)A universal relationc)An equivalence relationd)An empty relationCorrect answer is option 'C'. Can you explain this answer? for Class 12 2025 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R isa)Non commutative relationb)A universal relationc)An equivalence relationd)An empty relationCorrect answer is option 'C'. Can you explain this answer? covers all topics & solutions for Class 12 2025 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let T be the set of all triangles in a plane with R is a relation in T given by R = {(T1, T2) : T1 is congruent to T2}. Then R isa)Non commutative relationb)A universal relationc)An equivalence relationd)An empty relationCorrect answer is option 'C'. Can you explain this answer?.
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