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By defenition, a partition of a set is a set of non-empty subsets of such that every element in is in exactly one of these subsets.
a. Let , where . It is not a partition of a set . Indeed, is a unique set that contains the real number and is a unique set that contains the real number . On the other hand, a real number 1 belong to both set and
b. Let , where . Then it is a partition of a set . Let be a real number.The floor function is defined to be the greatest integer less than or equal to the real number . Let . Then . Since and are disjoint for different integers and , is a partition.