# Re[x] vs. x ∈ Reals in the context of Assuming

Let's say that $x$ is some real valued number $>0$.
Are the following commands interchangeable in the context of using Assuming?

1. Assuming[{Re[x] > 0}, Integrate[...,x]]
2. Assuming[{x ∈ Reals && x > 0}, Integrate[...,x]]

Or, as I suspect, is this not true for the reason that Re[x] > 0 should just mean that the Real component of $x$ is $>0$? Also, is there a way to more compactly specify {x ∈ Reals && x > 0}?

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No. Re[x] > 0 means that the real part of x is positive, but it does not mean that the imaginary part is zero. Re[1+I] > 0 but 1+I is not real.
However, x > 0 is sufficient and (in Mathematica) implies that x is also real.
@user11959 Not Norm, but Abs, in this case. The idea is that comparisons don't make sense for complex number. As soon as you use a comparison, Mathematica assumed that the associated variable is real. Yes, this is a peculiarity of Mathematica that you need to be aware of and not something immediately obvious. – Szabolcs Jan 26 '14 at 1:24