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

Check the explanation

Step-by-step explanation:

1

a) A is an Equivalence Relation

Reflexive : x is parallel to itself => x R x

Symmetric : x is parallel to y => y is parallel to x.

Therefore x R y => y R x

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 equivalent.

1. b)

x r y if and only if |x-y| less than or equal to 7

Reflexive : |x-x| = 0 <= 7 => x R x Satisfied.

Symmetric : let x R y => |x-y| <= 7

Consider |y-x| = |(-1)*(x-y)| = |x-y| <= 7

=> y R x => Satisfied

Transitive : let x R y and y R x

=> |x-y| <= 7 and |y-z| <= 7

but this doesn't imply x R z

Counter-Example : x = 1, y = 7, z = 10

Therefore this relation is neither Equivalent nor Partial Order Relation.