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 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.