Kate Kneale 's reply:
While there is a skill cross over you will be competing with people who have studied this specialist area so you will need to convince people this is actually what you want to do, not just an option because you haven't landed the job you wanted.
Set design is too vague. If I was interviewing you I would want to know exactly what you want to do: design for animation, tv, film, theatre, opera, dance? I would expect you to tell me what aspect of the specific artform you want to work in and why. Performance, narrative, lighting, different working process might be part of the reason.
Many people will have studied costume and set design before specialising - you should have an understanding of how costume is being used in these artforms even if it is not what you want to do. In low budget productions you might do both.
I would expect to see you going to lots of stuff. You need to build up your portfolio to demonstrate your interest. Volunteer, work on student projects, get drawing your own projects. Do an MA?
Good luck - hope this helps