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Question
Discuss the following relation for reflexivity, symmetricity and transitivity:
Let P denote the set of all straight lines in a plane. The relation R defined by “lRm if l is perpendicular to m”
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Solution
Let P denote the set of all straight lines in a plane.
The relation R is defined by l R m if l is perpendicular to m.
R = {(l, m): l is perpendicular to m}
(a) Reflexive:
Let l be any line in the plane P.
Then line l is not perpendicular to itself.
{1, 1) ∉ R
∴ R is not reflexive.
(b) Symmetric:
Let (1, m) ∉ R ⇒ l is perpendicular to m
∴ m is perpendicular to l.
Hence (m, l) ∈ R
∴ R is symmetric.
(c) Transitive:
Let (l, m), (m, n) ∈ R
⇒ l is perpendicular to m.
∴ l is parallel to n. (l, n) ∉ R
Hence R is not transitive.
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