In 9-33, determine whether the given relation is reflexive, symmetric, transitive, or none of these. Justify your answer.
Let A be the “punctured plane,”; that is, A is the set of all points in the Cartesian plane except the origin (0,0). A relation R is defined on A as follows: Fro every
To justify whether the given relation is reflexive, symmetric, transitive, or none of these.
Given information:
Let A be the “punctured plane”; that is, A is the set of all points in the Cartesian plane except the origin (0, 0). A relation R is defined on A as follows: For all p1 and p2 in A, p1R p2 ⇔ p1 and p2 lie on the same half line emanating from the origin.
Calculation:
Reflexive:
The relation R is reflective if
Since a point always lies on the same half line starting from the origin as itself, we know that
Symmetric:
The relation R on a set A is symmetric if
Let us assume that
By definition of R: