Define what it means for $\langle A,R\rangle$ to be a wellordering

Question

The definition of a wellordering is as follows:

If $\prec$ is a partial ordering of a set $X$.

$(X,\prec)$ is a wellordering if

(i) it is a strict total ordering and

(ii) for any subset $Y\subseteq X$, if $Y\neq\varnothing$, then $Y$ has a $\prec$-least element

How is defining $\langle A,R\rangle$ to be a wellordering different to the above definition,

Do these ordered pair brackets change the definition at all, if not why are they there?

Answer 1

The difference is typographical. Somewhat like the difference between $\Bbb R$ and $\bf R$ as the real numbers.

Sometimes people prefer using chevrons for ordered pairs, and sometimes parenthesis. It depends on the context, and the writer.

Answer 2

Despite the confusing symbology, there is no difference.

Simply:

• Read round brackets for the angled ones;
• Take it that the author uses $A$ as a standard letter for a set in place of $X$;
• Similarly, take it that the author uses $R$ to describe a relation instead of the suggestive $\prec$.

However, do note that depending on what $R$ is defined to be, you might have to add the additional clause that it is actually a partial ordering.