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 p 1 and p 2 in A , p 1 R p 2 ⇔ p 1 and p 2 lie on the same half line emanating from the origin.

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:

A= R×R

R={(x1,y1),(x2,y2)∈A|(x1,y1) and (x2,y2) lie on the same half line starting from the origin}

Reflexive:

The relation R is reflective if ((x1,y1),(x1,y1))∈R for every element (x1,y1)∈A.

Since a point always lies on the same half line starting from the origin as itself, we know that

((x1,y1),(x1,y1))∈R for all elements (x1,y1)∈A. Thus R is reflexive.

Symmetric:

The relation R on a set A is symmetric if ((x1,y1),(x2,y2))∈R whenever ((x2,y2),(x1,y1))∈R

Let us assume that ((x1,y1),(x2,y2))∈R.

By definition of R: (x1,y1) and (x2,y2) lie on the same half-line emanating from the origin, but then (x2,y2) and (x1,y1) also lie on the same half-line emanating from the origin