Final answer:
The questions involve determining whether sets form partitions and exploring mutual exclusivity and independence in set theory. Without details for set D, a definitive answer cannot be provided for partitions. Mutually exclusive sets cannot occur simultaneously, while independent sets have occurrences unaffected by each other.
Step-by-step explanation:
The questions posed involve concepts from set theory within the field of Mathematics. Given sets A, B, C, D, and E, we are asked to determine whether certain sets form partitions of another set and to explore the relationships between these sets in terms of mutual exclusivity and independence.
A partition of a set D means that:
- The sets A, B, C are non-empty,
- They are pairwise mutually exclusive (A AND B = empty set, A AND C = empty set, B AND C = empty set), and
- Their union forms the entire set D (A OR B OR C = D).
Without further information on set D, we cannot conclusively answer whether A, B, and C form a partition of D. However, if we had the specific elements of D, we could apply the definition of a partition to determine the answer.
Two events (or sets) are mutually exclusive if they cannot occur at the same time (or have no elements in common). They are independent if the occurrence of one does not affect the probability of the occurrence of the other. For instance, if set C represents an event with odd outcomes and set D represents an event with even outcomes smaller than five, C and D are mutually exclusive because they have no elements in common.
An event E, representing all outcomes less than five, is not mutually exclusive with event C because both sets have elements in common (the numbers 3 and 4). To determine independence, we need to consider if the probability of one event is unchanged by the occurrence of the other, which generally requires knowledge of the context or the ability to calculate probabilities.