Based on the discussion in Section 10.4.2, give a precise de?nition of when an access path covers the use of projection, union, and set-di?erence operators.

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1. Consider a relational expression of the form ?attr1,...,attrn(R).
The use of projection operator is covered by an access path if and only if one of the following conditions holds:
a. The access path is ?le scan.
b. The access path is a hash index whose search key is a subset of the attributes attr1, ...,attrn.
c. The access path is a B+ tree index with the search key sk1,...,skm and some pre?x sk1,...,ski of that search key is a subset of attr1,...,attrn.
d. The access path is binary search and the relation R is sorted on the attributes sk1,...,skm. The de?nition of “covering” in this case is the same as for B+ tree indices.
2. Consider relational expressions of the form R ? S or R ? S. The use of union or set-di?erence is covered by an access path if and only if one of the following conditions holds:
a. The access path is ?le scan.
b. The access path is a hash index on R and S.
c. The access path is a B+ tree index on R and S.
d. The access path is binary search and the relations R and S are sorted.

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