Partial order relation aRb is true when b contains a, 001R0101 is true - why? Note: I'm taking a course in Polish, where the word "ciąg" is used, which translates to "sequence", but may be translated as a "word" or a "string".

We have a set of binary sequences: A = { 0, 00, 01, 10, 11, 000, 001, 010, 100, 111, ... }.

We define a partial order relation $R \subseteq A^2$ so that $aRb$ is true if, and only if, sequence $b$ contains subsequence $a$. Examples: $001R0101$ is true, $110R0101$ is false.

...but 0101 doesn't contain 001 "as is", but rather after "skipping" the first 1 from it. I thought that if if a sequence contains a subsequence, it must contain it "in one block, as is" - without manipulating $b$ in any way. Why is this true/allowed?
In English math and computer science, there is a common notion of "subsequence" (not "substring") that does not have to be a contiguous block. Under this notion, if we have a string, say $a_1 a_2 a_3 \ldots a_n$, a subsequence is defined to be $a_{i_1} a_{i_2} a_{i_3} \ldots a_{i_k}$, where $1 \le i_1 &lt; i_2 &lt; \cdots &lt; i_k \le n$.
I would suspect that this is the definition intended in the course you are taking, and its a perfectly reasonable and useful definition. For, under this definition, $001$ is absolutely a subsequence of $0101$.

Note that the word "substring" is usually reserved for contiguous blocks, where "subsequence" may not have to be contiguous. This usage agrees with Wikipedia, see substring and subsequence.

Caveat: Of course, this is only speculation. This question is technically impossible to answer, because it's a matter of the way your course defined things. If your course makes a mistake, then how are we to know? And if not, there is nothing to answer definitively, only speculation. So I would avoid asking this type of question on mathSE, i.e. a question that does not have a real answer.
There must be a typo somewhere,
because it looks to me
that it should be false,
not true.