Which properties do each of the following relations satisfy among the properties relexive, symmetric transitiv, irreflexive, and antisymmetric?

1. R = {(a,b) l a squared = b squared} over the real numbers

2. R = {(x,y) l x divides y} over the positive integers

For each of the following conditions, find the smallest relation over the set A = {a,b,c} that satisfies the stated properties.

1. symmetric, not reflexive, not transitive

2. reflexive and symmetric, but not transitive

For each of the following properties, find a binary relation R such that R has the property but R squared does not.

1. Antisymmetric

Given the relation "less" over the natural numbers N, descre each of he following compositions as a set of the form {(x,y) l property}.

1. less less less