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The title is self-explanatory.

Let $a=1$, $b=2$

Is there a property we can state when we go from $1<2$ to $1 ≤2$?

Is there a property we can state when we go from $2=2$ to $2≤2$?

Can I say something like universal generalization or something?

This question stems from the fact that you can't go from $a≤b$ to $a<b$ but can go from $a<b$ to $a≤b$ so I assumed there is a name

Leon
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1 Answers1

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You could say something along the lines of one binary relation being "coarser" (weaker?) than the other. Thinking of $<, \leq, =$ as a subsets of $\mathbb{Z}^2$ (i.e., all those pairs where those relations are true), what you're stating is $=\subseteq \leq, <\subseteq \leq$. But now you can ask the same question about $\subset, \subseteq, =$ relations between subsets.

Also it comes from the definition of $\leq $being $<$ or $=$ (this is the same as the subset notion above).

red whisker
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  • So by saying "$\leq$" is weaker, that is a form of generalization right? – Leon Oct 31 '20 at 06:38
  • @Leon yes, similar notions exist when comparing topologies, $\sigma$-algebras etc. I think you should say $\leq$ is stronger/finer than $<$ because $\leq$ is true whenever $<$ is, but you could also say its weaker in some sense because $<$ is a strict inequality and $\leq$ is called a weak inequality (I borrowed the "stronger/finer" language from topology). – red whisker Oct 31 '20 at 07:07
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    I think another name is saying that a bound is sharp/sharper. –  Oct 31 '20 at 15:50