Tell whether the relation is reflexive, symmetric, asymmetric, antisymmetric or transitive.

Question

Tell whether the relation is reflexive, symmetric, asymmetric, antisymmetric or transitive.Identify equivalence relations or partial orders.

$R$ is the relation on people such that $a R b$ if $a$ and $b$ were born in the same year.

So I have an idea of the meaning of each of the terms symmetric, asymmetric etc but I am only familiar with how to test these using a matrix. How do I see if the relation is reflexive etc without making a matrix..?

Answer 1

You could read the definition of a binary relation in Wikipedia. Example 1.2 gives a good idea of how you should work with relations in general. The section "Relations over a Set" explain what a reflexive, symmetric, etc... relations are with the correct definitions.

So, how do you verify, for example, that your relation $R$ is reflexive? In formal terms, you have to verify that $aRa$, which means that $a$ is born in the same year as $a$, which is trivially true, right? If you have to write down a solution, I'd suggest writing something like "For all $a$, it is clear that $a$ is born in the same year as $a$, so $aRa$. Therefore, $R$ is reflexive".

Now, let's check that $R$ is not antisymmetric. Antisymmetry is the property that "$aRb$ and $bRa$ implies $a=b$", or, in terms of your relation, if you have two people $a$ and $b$ such that $a$ is born in the same year as $b$ and $b$ is born in the same year as $a$, then $a=b$. Simply take two different people $a$ and $b$ which are born in the same year. Then $aRb$ and $bRa$ but $a\neq b$, so $R$ is not antisymmetric.

To verify symmmetry, you have to prove or disprove that, if $a$ is born in the same year as $b$, then $b$ is born in the same year as $a$. What do you think? I'm going to leave the rest for you to solve by yourself.

Answer 2

You should ask yourself:

(1) Is it true for every person $x$ that $x$ was born in the same year as $x$ him- or her-self? (Reflexive).

(2) Is it true for all people $x,y$, if $x$ was born in the same year as $y$ does it necessarily follow that $y$ was born in the same year as $x$? (Symmetric).

See if you can take it from there and figure out transitive and the others.