Can I trasport a small set of tools with me in the airplane?

EU Regulations

The list of prohibited items drawn up by the EU doesn't mention screwdriver bits explicitly. I would not take this to mean that screwdriver bits are indeed allowed in your hand-luggage. Rather, these regulations are often left vague enough so as to allow the airport security staff enough wiggle room to rule on a case-by-case fashion. However, the linked document has an entire paragraph on workmen's tools which includes all of the following in the list of items forbidden in hand-luggage:

d. workmen's tools (tools capable of being used either to cause serious injury or to threaten the safety of aircraft), including:

• crowbars,
• drills and drill bits, including cordless portable power drills,
• tools with a blade or a shaft of more than 6 cm capable of use as a weapon, such as screwdrivers and chisels,
• saws, including cordless portable power saws,
• blowtorches,
• bolt guns and nail guns;

An over-zealous airport security staff member might consider as either screwdrivers, drill bits or both, and thus might disallow you from carrying them on the plane in your hand-luggage. As usual, to be safe, I'd put these in my check-in luggage.

TSA and CATSA Regulations

For completeness sake, note that TSA allows screwdrivers of length ≤ 7in to be carried in hand-luggage:

Similarly, the Canadian Air Transport Security Authority (CATSA) seems to allow tools of length ≤ 6cm in hand-luggage:

Once again, if the value of your screwdriver bits is too high to risk having them confiscated at the airport, I would check them in regardless of what the regulations say. Better be safe than sorry.

I was given a small set of screw drivers. A set of 8 bits and a plastic cable.
Can I transport that in hand-luggage?
I will be flying inside European Union.

Not with any certainty of their not being confiscated, regardless of what rules may or may not say.

I have had more innocuous items than that confiscated (not in Europe).

FWIW, if it was in a pack pocket surrounded by coins and other small miscellaneous metal paraphenalia then a medium size Swiss Army Knife might make the trip from London to Paris undetected.(Or may not).
(As may happen if a passenger discovered that they were carrying it only after their main baggage had been checked*.)(Don't ask me how I know). That being the case, the same may apply to a very small toolkit. If this did happen the passenger would probably want their awareness of the tools inclusion to be plausibly deniable.

• Said hypothetical knife may have (hypothetically) been the one which fell foul of a Chinese inland airline security check about 10 years later. Doh!