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Consider the following statement.
I. Parallelism of lines is an equivalence relation.
II. xRy, if x is a father of y, is an equivalence relation. Which of the statements given above is/are correct?
I. Parallelism of lines is an equivalence relation.
II. xRy, if x is a father of y, is an equivalence relation. Which of the statements given above is/are correct?
- a)I only
- b)II only
- c)Both I and II
- d)Neither I nor II
Correct answer is option 'A'. Can you explain this answer?
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Verified Answer
Consider the following statement.I. Parallelism of lines is an equival...
I. Let l, m, n are parallel lines and R is a relation.
Then l|| l, then R is reflexive,
and l || m and m || l, then R is symmetric.
Also, l || m, m || n => l | n, then R is transitive. Hence, R is an equivalence relation.
II. If x is father of y and y is not father of x then relation is not symmetric, thus relation is not equivalence
Then l|| l, then R is reflexive,
and l || m and m || l, then R is symmetric.
Also, l || m, m || n => l | n, then R is transitive. Hence, R is an equivalence relation.
II. If x is father of y and y is not father of x then relation is not symmetric, thus relation is not equivalence
Most Upvoted Answer
Consider the following statement.I. Parallelism of lines is an equival...
I. Let l, m, n are parallel lines and R is a relation.
Then l|| l, then R is reflexive,
and l || m and m || l, then R is symmetric.
Also, l || m, m || n => l | n, then R is transitive. Hence, R is an equivalence relation.
II. If x is father of y and y is not father of x then relation is not symmetric, thus relation is not equivalence
Then l|| l, then R is reflexive,
and l || m and m || l, then R is symmetric.
Also, l || m, m || n => l | n, then R is transitive. Hence, R is an equivalence relation.
II. If x is father of y and y is not father of x then relation is not symmetric, thus relation is not equivalence
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Consider the following statement.I. Parallelism of lines is an equival...
Understanding Equivalence Relations
An equivalence relation is a relation that satisfies three properties: reflexivity, symmetry, and transitivity. Let's analyze the two statements to see which qualifies as an equivalence relation.
Statement I: Parallelism of Lines
- Reflexivity: Any line is parallel to itself. This holds true.
- Symmetry: If line A is parallel to line B, then line B is parallel to line A. This is also satisfied.
- Transitivity: If line A is parallel to line B and line B is parallel to line C, then line A is parallel to line C. This holds true as well.
Since all three properties are satisfied, parallelism of lines is indeed an equivalence relation.
Statement II: xRy, if x is a father of y
- Reflexivity: A father cannot be his own child; thus, x cannot be related to itself.
- Symmetry: If x is a father of y, it does not imply that y is a father of x. This fails.
- Transitivity: Even if x is a father of y and y is a father of z, x is not a father of z.
Since it fails to satisfy reflexivity and symmetry, the relation "x is a father of y" is not an equivalence relation.
Conclusion
As a result, the correct answer is option 'A': only Statement I is correct, confirming that parallelism of lines is an equivalence relation while the father-child relationship is not.
An equivalence relation is a relation that satisfies three properties: reflexivity, symmetry, and transitivity. Let's analyze the two statements to see which qualifies as an equivalence relation.
Statement I: Parallelism of Lines
- Reflexivity: Any line is parallel to itself. This holds true.
- Symmetry: If line A is parallel to line B, then line B is parallel to line A. This is also satisfied.
- Transitivity: If line A is parallel to line B and line B is parallel to line C, then line A is parallel to line C. This holds true as well.
Since all three properties are satisfied, parallelism of lines is indeed an equivalence relation.
Statement II: xRy, if x is a father of y
- Reflexivity: A father cannot be his own child; thus, x cannot be related to itself.
- Symmetry: If x is a father of y, it does not imply that y is a father of x. This fails.
- Transitivity: Even if x is a father of y and y is a father of z, x is not a father of z.
Since it fails to satisfy reflexivity and symmetry, the relation "x is a father of y" is not an equivalence relation.
Conclusion
As a result, the correct answer is option 'A': only Statement I is correct, confirming that parallelism of lines is an equivalence relation while the father-child relationship is not.
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