Rationals subset of reals?

When constructing the real numbers R from the rational numbers Q, is it really formally legitimate to say that Q is a subset of R? I have strong doubts about this, because R is usually defined in terms of equivalence classes of sets of rational numbers. So what's meant by the symbol Q actually? The set of rational numbers themselves (i.e., certain equivalence classes of pairs of integers) or the subset of R which corresponds to these rational numbers?

Thanks very much in advance.
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