comparison with logic problems

Question

I am a bit confused regarding the logic combined with size comparisons. For example, if there is a statement

x > y -> x >= y

I believe that this would be true, as the former is a subset of the latter. However, would the statement

x >=y -> x > y

also be considered true? I believe in this case, the statement would be false.

To explain this, I was thinking about the combination of "or" in the if statement that is: if there exists a or in the if statement

(x > y or x = y)

the conclusion needs to satisfy both conditions in the or statement

x = y is not included in x > y

Would this be the correct understanding of logical comparisons?

Answer 1

In logic:
$x\geq y \iff x>y \lor x=y$ is true.

But, $x\geq y\implies x>y$ is not a valid inference, because:
$p\lor q$ does not tautologically imply $p$

So your naive set approach is correct reasoning .