Let L be the set of all lines in a plane and R be the relation in L defined as R={(L1, L2):L1 is perpendicular to L2}. Show that R is symmetric but neither reflexive nor transitive.
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Step-by-Step Solution
Step 1: Check for Reflexivity
For a relation to be reflexive, every element must be related to itself. In this case, a line L1 would need to be perpendicular to itself. However, a line cannot be perpendicular to itself, as the angle between a line and itself is 0∘, not 90∘. Therefore, the relation R is not reflexive.
Step 2: Check for Symmetry
For a relation to be symmetric, if L1 is related to L2, then L2 must be related to L1. If line L1 is perpendicular to line L2, it implies that the angle between them is 90∘. This also means that line L2 is perpendicular to line L1. Therefore, the relation R is symmetric.
Step 3: Check for Transitivity
For a relation to be transitive, if L1 is related to L2 and L2 is related to L3, then L1 must be related to L3. If L1 is perpendicular to L2, and L2 is perpendicular to L3, then L1 and L3 must be parallel to each other (or coincident). Parallel lines are not perpendicular. Therefore, the relation R is not transitive.