I am working on a homework assignment for a discrete math course and am completely lost on relations. I'll put up some examples of problems, could somebody please push me in the right direction or explain the answers? Thank you!
For all of these you can pick one or more answers, any input would be greatly appreciated.
$1$. Define a relation $∼$ on $\mathbb{Z}$ by $x∼y \iff xy=1$. This relation has which of the following properties?
a. Reflexive
b. Irreflexive
c. Symmetric
d. Antisymmetric
e. Transitive
$2$. Let R be a relation over all integers, so that for any two integers $x,y$, we have $x$R$y$ $\iff$ $|x|=|y|$. Then, choose all that hold:
a. R is a total ordering
b. R is irreflexive
c. R is anti-symmetric
d. R is symmetric
e. R is a partial ordering
f. R is transitive
g. R is an equivalence relation
h. R is reflexive
$3$. Consider the divides relation (p∣q) on the set of integers. What properties does this relation have?
a. irreflexive
b. symmetric
c. reflexive
d. antisymmetric
e. transitive
$4$. Let's define a relation R on R2 as follows $(x,y)$ R $(p,q)$ $\iff$ $x^2+y^2<p^2+q^2$ Which properties does this relation have?
a. transitive
b. antisymmetric
c. reflexive
d. irreflexive
e. symmetric
Reflexive: A binary relation $R$ is reflexive if $a\ R\ a$ for every $a$.
(1) Is it true that $xx = 1$ for all $x \in \Bbb Z$?
(2) Is it true that $|x| = |x|$ for all $x \in \Bbb Z$?
(3) Is it true that $p\mid p$ for all $p \in \Bbb Z$?
(4) Is it true that $x^2 + y^2 = x^2 + y^2$ for every $(x,y) \in \Bbb R^2$?
Irreflexive: A binary relation $R$ is irreflexive if $a\ R\ a$ is never true for any $a$.
Symmetric: A binary relation $R$ is symmetric if $b\ R\ a$ is true whenever $a\ R\ b$ is true.
(1) If $xy = 1$, is it always true that $yx = 1$?
(2) if $|x| = |y|$, is it always true that $|y| = |x|$?
(3) if $p\mid q$, is it always true that $q\mid p$?
(4) if $x^2 + y^2 = p^2 + q^2$, is it always true that $p^2 + q^2 = x^2 + y^2$?
Antisymmetric: $R$ is antisymmetric if whenever $a\ R\ b$ and $a \ne b$, it is false that $b\ R\ a$. (Some definitions may not include the restriction that $a \ne b$ - check your textbook or notes to find out what definition you use. I leave it in so that $\le$ and $\ge$ qualify.)
Transitive: $R$ is transitive if whenever $a\ R\ b$ and $b\ R\ c$, we also have $a\ R\ c$.
(1) If $xy = 1$ and $yz = 1$, does it follow that $xz = 1$?
(2) if $|x| = |y|$ and $|y| = |z|$, does it follow that $|x| = |z|$?
(3) if $p\mid q$ and $q\mid r$, does it follow that $p\mid r$?
(4) if $x^2 + y^2 = p^2 + q^2$ and $p^2 + q^2 = u^2 + v^2$, does it follow that $x^2 + y^2 = u^2 + v^2$?
Equivalence relation: $R$ is an equivalence relation if it is reflexive, symmetric, and transitive.
Partial order: $R$ is a partial order if it is (either reflexive or irreflexive), antisymmetric, and transitive. (Again, your book may differ as to whether reflexive or irreflexive is required, or if either one will work. Irreflexive partial orders include $<$ and $>$. Reflexive partial orders include $\le$ and $\ge$.)
Total order: $R$ is a total order if it is a partial order, and satifies that for all $a \ne b$, either $a\ R\ b$ or $b\ R\ a$ holds.