They are not mutually exclusive because they are defined in orthogonal ways. "Vegetable" is defined by how a plant is cooked and eaten, and so has a lot of cul

They are not mutually exclusive because they are defined in orthogonal ways. "Vegetable" is defined by how a plant is cooked and eaten, and so has a lot of cultural and context specific influences. A vegetable can be a root (potato), a stalk (celery), a fruit (cucumber), seeds (beans), even non-plant fungi (mushroom). A fruit has a very specific definition (especially in a biological or botanical usage). It's the "seed-bearing structure in flowering plants that is formed from the ovary after flowering" (Wikipedia). The part of the cucumber we eat is the fruit (you can see its seeds). But the way we prepare and eat it means we classify it as a vegetable.