(a) In law, a business that is run by a sole trader is not treated as separate in any way from its owner. The sole trader is personally responsible for anything they do in the course of running their business.
(b) In law, a sole trader's business is treated as a separate entity, distinct from the owner of that business. Whatever a sole trader does in the course of running the business does not make the owner personally responsible: it is the business itself that is legally bound.
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