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Youtube Clip: Quotient Groups Part 2
The video shows that "the operation on quotient group is well-defined" by taking two elements from two different cosets and showing their composition leads to some element in one of the other cosets but I don't understand the reason why it needs to be shown since any element is in one of the cosets including $a \circ b$ (cosets partitions the group). I know that to show a function is well-defined I need to show every element can be mapped to exactly one element. But how about this. Are they conceptually the same?