Definition: $a \prec b$ , $ a \npreceq b$ , $a,b \in \mathbb{R}$

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let be $a,b \in \mathbb{R}$:

$a \prec b$ if $a \preceq b \wedge a \neq b$

$a \npreceq b $ if $b \prec a$

is correct?

Thanks in advance!

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Yes, it is correct. For more information check this website