Understanding Irreflexivity in Relations?

Question

I'm having difficulty understanding irreflexive relations.

I have the definition of an irreflexive relation as: a relation $R$ over a set $A$ if for any $x \in A$, $xRx$ doesn't hold.

Many sources indicate that the relation "less than", for example, is irreflexive in this way ($x<x$ doesn't hold).

My question is: why not? Isn't $x<x$ simply "false"? I may be misunderstanding the meaning of "$xRx$ holds (or doesn't hold)".

For context, my background is in software development, which may be clouding my intuition. It's not as though the expression $2<2$ wouldn't compile, for example; it would just return false, but it seems this isn't mathematically accurate.

Answer 1

A relation on $A$ is formally a collection of ordered pairs of elements $(a,b)$. Generally when we work with a specific example of a relation, we are told a description or definition of our specific relation. For specific example, the relation $\prec$ on the set $\{1,2,3,4\}$ where a pair $(a,b)$ is included in our relation if and only if $a$ is "less than" $b$.

Formally for natural numbers, we can say $a<b$ iff there exists some positive $n$ such that $S^n(a)=b$ where $S(\cdot)$ is the successor function.

We have in this specific example $\prec~=\{(1,2),(1,3),(1,4),(2,3),(2,4),(3,4)\}$

For shorthand we will say $a\prec b$ to mean that $(a,b)\in ~\prec$.

Further, we will say $a\not\prec b$ to mean that $(a,b)\notin~\prec$.

In this example $2\not\prec 2$ is a true statement. $2\prec 2$ is a false statement. To be irreflexive we need $x\not\prec x$ to be true for all $x$. This is the same as $x\prec x$ to be false for all $x$. Worded a third way, that is to say that in the collection of pairs in our relation, there exists no pair of the form $(x,x)$. Our example above is indeed irreflexive.

Answer 2

Doesn't hold literally means false. It means that the statement that x is related to x under relation '<' is false, because x is not related to x under the relation '<'.

"if an element x is not related to another element y under relation R, then xRy is false(or Doesn't hold)."