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If I know for two numbers $a$ and $b$ that

$${a < b }$$

Then is it correct to say that

$$ a \leq b $$

I know that the second statement is true as long as the first one is.

However is it true that

$$a \leq b \implies a<b$$

That is does less than or equal imply less than?

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    $\begingroup$ The header question only matches one of the question in the body of your post. It is true that $a<b\implies a≤b$ . It is not true that $a≤b\implies a<b$. $\endgroup$ – lulu Feb 8 at 23:55
  • $\begingroup$ @lulu The header question is precisely the question in the end of the post? $\endgroup$ – mrtaurho Feb 8 at 23:56
  • $\begingroup$ @mrtaurho True. I guess both directions are asked in the body of the post. I will edit my first comment accordingly. $\endgroup$ – lulu Feb 8 at 23:57
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You're correct in the first case.

But now consider the proposition $a\leq b.$ In this case if $a=b=4$, $a\leq b$ is true.

However given $a< b$, it is false that $4<4$. So it is false that $a\leq b \to a\lt b$ is always true.

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$1\leq 1$ (since $1=1$) but does not hold that $1<1$.

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No. Take $a=b$ to see that this does not hold in general. Clearly, $a\le a$ but surely not $a<a$.

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No.

Take any number $a$ in your domain. Then $a=a$ (and so $a\le a$) but $a\not<a$.

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No. $a\le b$ means "a less than or equal to b." It could be equal to, hence not $a<b$.

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