Actually your element does not depend on your astrological sign because all people have all signs and your planets and calculated points will be in several of them.
People also do not "have an element.". You may be gifted with a particular element, but as different elements do different things it would be foolish to try and specialize in one (or two, as the case may be).
What people think of as "your sign" is basically ust your sun sign; you also have a moon sign, a mercury sign, a Venus sign, etc.. Your natal astrology is a snapshot of where everything in the astrological sky is, not just your sun. For example, I'm sun and Venus Aries, Luna Virgo, Mercuy and Jupiter Pisces, etc. The signs themselves, even those you don't have a planet it, still affect you and this can be seen most readily in the Rising sign and the Midheaven.
Humans are made up of and contain all elements. Each element has its own, particular use.