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 Z by x∼y if and only if 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 xRy 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) if and only if x2+y2

a. transitive

b. antisymmetric

c. reflexive

d. irreflexive

e. symmetric


closed as off-topic by Ian Miller, Leucippus, user228113, hardmath, user296602 Feb 24 '16 at 4:59

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    $\begingroup$ You need to explain what you don't understand. Do you understand what it means to say a relation is symmetric, reflexive, etc.? $\endgroup$ – symplectomorphic Feb 24 '16 at 0:21
  • $\begingroup$ Sorry this is my first post, was not sure what to put, yes I do not understand at all what it means for a relation to be symmetric, reflexive, etc. Especially when having to do with functions. $\endgroup$ – Gonez Feb 24 '16 at 1:24
  • $\begingroup$ Definitions are your friends. $\endgroup$ – hardmath Feb 24 '16 at 2:55

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 $a$.

  • are any of the statements above FALSE for every $x,p$, or $(x,y)$?

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.)

  • for any of the conditions above, is it true that when the first holds for unequal values, the second is never true?

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.

  • Did any of the 4 relations pass these three tests?

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 $\ge$.)

  • Are any of the 4 relations transitive and antisymmetric? Of those, are they either reflexive or irreflexive?

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.

  • Of the partial orders, are there any two values in the set on which they are defined, for which the relation does not hold in either order?
  • $\begingroup$ Thank you very much, that will help a lot for my exam tomorrow. $\endgroup$ – Gonez Feb 24 '16 at 22:48

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