Question

4. Determine which of the following are equivalence relations and/or partial CHAPTER 6. RELATIONS 121 ordering relations for the given sets: (a) A = { lines in the plane}, and r defined by xry if and only if x is parallel to y. Assume every line is parallel to itself. (b) A = R and r defined by xry if and only if |x − y| ≤ 7.

Answer 1

Determine which of the following are equivalence relations and/or partial CHAPTER 6. RELATIONS 121 ordering relations for the given sets: (a) A = { lines in the plane}, and r defined by xry if and only if x is parallel to y. Assume every line is parallel to itself. (b) A = R and r defined by xry if and only if |x − y| ≤ 7.

A is an Equivalence Relation.

Explanation:

For a reflexive case : x is parallel to itself => x R x

For a symmetric : x is parallel to y => y is parallel to x.

Thus, x R y = y R x

For a transitive: x is parallel to y, and y is parallel to z then x, y, z are parallel to each other.

=> x R y and y R z = x R z

Therefore, A is an Equivalence Relation.