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Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2):I1 is parallel to I2}. What is the type of given relation?
- a)Reflexive relation
- b)Transitive relation
- c)Symmetric relation
- d)Equivalence relation
Correct answer is option 'D'. Can you explain this answer?
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Most Upvoted Answer
Let I be a set of all lines in a XY plane and R be a relation in I def...
Explanation:
To determine the type of the given relation, let's analyze the properties of the relation R.
Reflexive relation:
A relation is reflexive if every element is related to itself. In this case, it means that every line is parallel to itself. However, this is not true since a line cannot be parallel to itself. Therefore, the given relation is not reflexive.
Symmetric relation:
A relation is symmetric if for every element (a, b) in the relation, (b, a) is also in the relation. In this case, it means that if line I1 is parallel to line I2, then line I2 is also parallel to line I1. This property holds true since the concept of parallel lines is symmetric. Therefore, the given relation is symmetric.
Transitive relation:
A relation is transitive if for every elements (a, b) and (b, c) in the relation, (a, c) is also in the relation. In this case, it means that if line I1 is parallel to line I2, and line I2 is parallel to line I3, then line I1 is also parallel to line I3. This property holds true since the transitive property of parallel lines states that if two lines are parallel to the same line, then they are parallel to each other. Therefore, the given relation is transitive.
Equivalence relation:
An equivalence relation is a relation that is reflexive, symmetric, and transitive. From the above analysis, we can see that the given relation is not reflexive but it is symmetric and transitive. Therefore, the given relation R is an equivalence relation.
Conclusion:
The given relation R = {(I1, I2): I1 is parallel to I2} is an equivalence relation.
To determine the type of the given relation, let's analyze the properties of the relation R.
Reflexive relation:
A relation is reflexive if every element is related to itself. In this case, it means that every line is parallel to itself. However, this is not true since a line cannot be parallel to itself. Therefore, the given relation is not reflexive.
Symmetric relation:
A relation is symmetric if for every element (a, b) in the relation, (b, a) is also in the relation. In this case, it means that if line I1 is parallel to line I2, then line I2 is also parallel to line I1. This property holds true since the concept of parallel lines is symmetric. Therefore, the given relation is symmetric.
Transitive relation:
A relation is transitive if for every elements (a, b) and (b, c) in the relation, (a, c) is also in the relation. In this case, it means that if line I1 is parallel to line I2, and line I2 is parallel to line I3, then line I1 is also parallel to line I3. This property holds true since the transitive property of parallel lines states that if two lines are parallel to the same line, then they are parallel to each other. Therefore, the given relation is transitive.
Equivalence relation:
An equivalence relation is a relation that is reflexive, symmetric, and transitive. From the above analysis, we can see that the given relation is not reflexive but it is symmetric and transitive. Therefore, the given relation R is an equivalence relation.
Conclusion:
The given relation R = {(I1, I2): I1 is parallel to I2} is an equivalence relation.
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Let I be a set of all lines in a XY plane and R be a relation in I def...
This is an equivalence relation. A relation R is said to be an equivalence relation when it is reflexive, transitive and symmetric.
Reflexive: We know that a line is always parallel to itself. This implies that I1 is parallel to I1 i.e. (I1, I2)∈R. Hence, it is a reflexive relation.
Symmetric: Now if a line I1 || I2 then the line I2 || I1. Therefore, (I1, I2)∈R implies that (I2, I1)∈R. Hence, it is a symmetric relation.
Transitive: If two lines (I1, I3) are parallel to a third line (I2) then they will be parallel to each other i.e. if (I1, I2) ∈R and (I2, I3) ∈R implies that (I1, I3) ∈R.
Reflexive: We know that a line is always parallel to itself. This implies that I1 is parallel to I1 i.e. (I1, I2)∈R. Hence, it is a reflexive relation.
Symmetric: Now if a line I1 || I2 then the line I2 || I1. Therefore, (I1, I2)∈R implies that (I2, I1)∈R. Hence, it is a symmetric relation.
Transitive: If two lines (I1, I3) are parallel to a third line (I2) then they will be parallel to each other i.e. if (I1, I2) ∈R and (I2, I3) ∈R implies that (I1, I3) ∈R.
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Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2):I1 is parallel to I2}. What is the type of given relation?a)Reflexive relationb)Transitive relationc)Symmetric relationd)Equivalence relationCorrect answer is option 'D'. Can you explain this answer?
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Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2):I1 is parallel to I2}. What is the type of given relation?a)Reflexive relationb)Transitive relationc)Symmetric relationd)Equivalence relationCorrect answer is option 'D'. Can you explain this answer? for Class 12 2024 is part of Class 12 preparation. The Question and answers have been prepared according to the Class 12 exam syllabus. Information about Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2):I1 is parallel to I2}. What is the type of given relation?a)Reflexive relationb)Transitive relationc)Symmetric relationd)Equivalence relationCorrect answer is option 'D'. Can you explain this answer? covers all topics & solutions for Class 12 2024 Exam. Find important definitions, questions, meanings, examples, exercises and tests below for Let I be a set of all lines in a XY plane and R be a relation in I defined as R = {(I1, I2):I1 is parallel to I2}. What is the type of given relation?a)Reflexive relationb)Transitive relationc)Symmetric relationd)Equivalence relationCorrect answer is option 'D'. Can you explain this answer?.
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