Answer:
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.
