# Thread: reflexive, symmetric, antisymmetric, transitive?

1. ## reflexive, symmetric, antisymmetric, transitive?

For each of these binary relations, determine whether they are reflexive, symmetric,
antisymmetric, transitive. Give reasons for your answers and state whether or not they
form order relations or equivalence relations.

On the set {audi, ford, bmw, mercedes}, the relation
{(audi, audi), (audi, bmw), (bmw, bmw), (ford, ford), (mercedes,mercedes),
(audi, mercedes), (audi, ford), (bmw, ford), (mercedes, ford) }.

Let F be the set of all possible filenames consisting of character strings of at
least one character. The relation R contains all pairs of names (name1, name2)
where the first eight characters of name1 are the same as the first eight
characters of name2, or if name1 and name 2 have fewer than eight characters
and are exactly the same.

2. What have you tried so far?

3. I don't really know were to start

4. Originally Posted by jander1
I don't really know were to start

Start with the reflexive. A binary relation $\displaystyle R$ on a set $\displaystyle S$ is said to be reflexive iff $\displaystyle aRa$ for all $\displaystyle a\in S$ .

Are those relations reflexive?

5. Originally Posted by jander1
I don't really know were to start

6. Suppose the parent set is $\displaystyle \{a_1,a_2,a_3,a_4\}$. A relation defined on this set is reflexive if the relation-set contains the element $\displaystyle (a_i,a_i)$ for all $\displaystyle i=1,2,3,4$. Is it the case here?

A relation defined on this set is symmetric if for any $\displaystyle (a_i,a_j)$ belonging to the relation-set, the element $\displaystyle (a_j,a_i)$ is also present in the relation-set. Is it the case here for all $\displaystyle i,j=1,2,3,4$?

A relation defined on this set is transitive if for two elements $\displaystyle (a_i,a_j)$ and $\displaystyle (a_j,a_k)$ belonging to the relation-set, the element $\displaystyle (a_i,a_k)$ is also present in the relation-set. Is it the case here for all $\displaystyle i,j,k=1,2,3,4?$

Proceed step by step; realize the concept and tell what you get.