Sketch-based streaming algorithms allow efficient processing of big data. These algorithms use small fixed-size storage to store a summary ("sketch") of the input data, and use probabilistic algorithms to estimate the desired quantity. However, in many real-world applications it is impractical to collect and process the entire data stream, the common practice is thus to sample and process only a small part of it. While sampling is crucial for handling massive data sets, it may reduce accuracy. In this paper we present a new framework that can accurately estimate the cardinality of any set expression between any number of streams using only a small sample of each stream. The proposed framework consists of a new sketch, called Maximal-Term with Subsample (MTS), and a family of algorithms that use this sketch. An example of a possible query that can be efficiently answered using the proposed sketch is, How many distinct tuples appear in tables $T_1$ and $T_2$, but not in $T_3$? The algorithms presented in this paper answer such queries accurately, processing only a small sample of the tuples in each table and using a constant amount of memory. Such estimations are useful for the optimization of queries over very large database systems. We show that all our algorithms are unbiased, and we analyze their asymptotic variance.