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

2. Hello, patches!

Here's some help . . .

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

$\displaystyle 1)\;\;R \:= \:\{(a,b)\:|\:a^2= b^2\}$ over the real numbers

$\displaystyle a^2\,=\,a^2$ . . . True: reflexive

$\displaystyle \text{If }a^2 \,=\,b^2\text{, then }b^2\,=\,a^2$ . . . True: symmetric

$\displaystyle \text{If }a^2\,=\,b^2\text{ and }b^2\,=\,c^2\text{, then }a^2\,=\,c^2$ . . . True: transitive

$\displaystyle 2)\;\;R \:= \: \{(x,y)\:|\:x\text{ divides }y\}$ over the positive integers

$\displaystyle a\:\boxed{D}\:a$ . . . True: reflexive

$\displaystyle \text{If }a\:\boxed{D}\:b\text{, then }b\:\boxed{D}\:a$ . . . False: not symmetric

$\displaystyle \text{If }a\:\boxed{D}\:b\text{ and }b\:\boxed{D}\:a\text{, then }a = b$ . . . True: antisymmetric

$\displaystyle \text{If }a\:\boxed{D}\:b\text{ and }b\:\boxed{D}\:c\text{, then }a\:\boxed{D}\:c$ . . . True: transitive

3. 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
1. $\displaystyle a \neq b$ or a nicer one, gcd(a,b) = 1;

2. this one's tricky, try gcd(a,b) = max(a,b)