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If S equals straight lines in the plane and ab if a and b are parallel Verify that the relation is an equivalence relation on the set S given?
Wiki User
∙ 14y ago
Establishing equivalence depends on the definition of parallel lines. If they are defined as lines which cannot ever meet (have no point in common), then the relation is not reflexive and so cannot be an equivalence relation.
However, if the lines are in a coordinate plane and parallel lines are defined as those which have the same gradient then:
- the gradient of a is the gradient of a so the relationship is reflexive ie a ~ a.
- if the gradient of a is m then b is parallel to a if gradient of b = m and, if the gradient of b is m then b is parallel to a. Thus the relation ship is symmetric ie a ~ b <=> b ~ a.
- If the gradient of a is m then b is parallel to a if and only if gradient of b = gradient of a, which is m. Also c is parallel to b if and only if gradient of c = gradient of b which is m. Therefore c is parallel to a. Thus the relation is transitive, that is a ~ b and b ~ c => a ~ c.
The relation is reflexive, symmetric and transitive and therefore it is an equivalence relationship.
Wiki User
∙ 6y ago
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