Usage of 'knows better' (I think I shouldn't go hiking but Grace tells me I...)

[Azz]

Is there anything wrong with these sentences:

a. I thought I should not turn off the computer but Kate knows better and she told me to turn it off.
b.I think I shouldn't go hiking but Grace tells me I should and she knows better.

In both cases 'knows better' is used instead of 'knows better than me'.

[Kitosdad]

Good morning Azz. Yes, it does.

Kitos.
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Englishman living in Germany.