No!
Despite what your grade school math teacher taught you, these equivalences
don't always work in software, especially with floating point expressions or
user-defined types.
Example: if a is a floating point NaN, then both a
< b and a >= b will be false. That means !(a < b) will be
true and a >= b will be false.
Example: if a is an object of class Foo that has overloaded
operator< and operator>=, then it is up to the creator of
class Foo if these operators will have opposite semantics. They
probably should have opposite semantics, but that's up to whoever
wrote class Foo.