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I came across this following problem:

Let $R$ be a reflexive relation on a finite set $A$ having $n$ elements and let there be $m$ ordered pairs in $R$, then:
A) $m\ge n$
B) $m \le n$
C) $m=n$
D) None of these

I initially started with taking a simple example:
Let set $A$ was ${1,2,3,4,5,6}$ then one of the relations on it could be:
$R={(1,1),(2,2),(1,2)}$ which is reflexive right?

so this(according to my knowledge) showed that m doesn't have to be $\le$ or $\ge$ than n. But the answer showed that I was wrong and $m\ge n$ But why does this happen?

Is it necessary for all the elements of $A$ to be included in $\mathbb R$ for it to be reflexive? That is, considering the previous example, do the pairs $(1,1), (2,2),(3,3),(4,4),(5,5),(6,6)$ need to exist in $R$ for it to be reflexive?
I have this conceptual confusion about what a reflexive relation really is and would be glad if someone could clear this out.

Another sort of counter question to this, then technically is an equivalence relation even possible for sets like the set of natural numbers as the following example illustrates what I mean:
lets take any relation $R$ defined on the set of natural numbers,

$R=${$(2,3),(3,4)$}

then what minimum number of ordered pairs should we add to make it an equivalence relation ?

Does the word 'minimum' allow us to not care about adding all the ordered pairs of natural numbers, that is, $(1,1),(2,2),(3,3),(4,4)$ and so on...?

Prajwal Tiwari
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    Look at the definition of reflexive... a relation $R$ over a set $X$ is called reflexive if and only if $(x,x)\in R$ for every $x\in X$. The relation ${(1,1),(2,2),(1,2)}$ over the set ${1,2,3,4,5,6}$ is not reflexive because it does not, for example, contain the pair $(3,3)$ among others. – JMoravitz Apr 30 '21 at 17:13
  • Worded another way, letting $\operatorname{Id}_X = {(x,x)~:~x\in X}$ be the identity relation (equivalently identity function) on $X$, a relation $R$ over $X$ is reflexive if and only $\operatorname{Id}_X\subseteq R$. Recognize that $|\operatorname{Id}_X|=|X|$ and so given $\operatorname{Id}_X\subseteq R$ we have $|X|\leq |R|$ – JMoravitz Apr 30 '21 at 17:15
  • @Algebrology do not typeset relations with blackboard font... $\Bbb R$ is exclusively used to refer to the set of real numbers. $R$ here in this problem is just used as a name for a relation... Also, escape curly braces to have them not disappear when typeset. $\{1,2,3\}$ produces ${1,2,3}$ while ${1,2,3}$ the braces disappear as ${1,2,3}$ – JMoravitz Apr 30 '21 at 17:16
  • @JMoravitz I have made another edit to my post which extends my question here... Kindly take a look at it – Prajwal Tiwari Apr 30 '21 at 17:23
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    One of the first math.stackexchange.com questions "Why isn't reflexivity redundant in the definition of equivalence relation?" is related. – Somos Apr 30 '21 at 17:24
  • "Is an equivalence relation possible for sets (that are infinite)" Yes, of course. "What minimum number of pairs we need to add for this example" Infinitely many because we were missing infinitely many reflexive pairs. Beyond the infinitely many reflexive pairs we were missing, we would also require at a minimum $(2,4),(3,2),(4,2),(4,3)$ – JMoravitz Apr 30 '21 at 17:28
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    To give a "real-world example": consider the relation $R$ on people, where $aRb$ if $a$ and $b$ are contemporaries, i.e., if their lifespans overlap. Then certainly this is reflexive, since everyone lives at the same time as themselves. You must have $aRa$ for every person $a$ in your list; no one will be omitted. – Théophile Apr 30 '21 at 17:35

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A relation $R$ on a set $A$ is reflexive iff for every $a\in A$ we have $(a,a)\in R$. Note that this means that unlike transitivity and symmetry, reflexivity crucially depends on the "carrier set" involved". So for example, $\{(1,1),(2,2),(1,2)\}$ is reflexive as a relation on $\{1,2\}$ but is not reflexive as a relation on $\{1,2,3\}$.

Note that one consequence is that a reflexive relation on a set must have at least as many elements as that set; in particular, addressing one of your later questions, every reflexive relation on $\mathbb{N}$ contains infinitely many ordered pairs.

This "context dependence" may feel odd at first, but it's a crucial aspect of the definition.

Noah Schweber
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