Final answer:
The smallest field containing sets A and B coincides with the field containing the partition {AB, A*Bᴅ, Aᴅ*B, Aᴅ*Bᴅ} because A and B can be reconstructed from the partition and vice versa.
Step-by-step explanation:
The question at hand asks us to demonstrate that the smallest field containing two sets, A and B, is the same as the field containing the partition {AB, A*Bᴅ, Aᴅ*B, Aᴅ*Bᴅ}. To show this, one must understand the concepts of fields in set theory and partitions. The partition mentioned is a collection of non-overlapping sets that together cover all the possibilities of the union of sets A and B. Clearly, the intersection AB and the other parts of the partition are subsets of the join of A and B, and their join is A union B. Thus, the smallest field containing the partition also contains A and B. Conversely, A and B can be expressed in terms of the partition, since A = AB ∪ (A*Bᴅ) and B = AB ∪ (Aᴅ*B), which means that the smallest field containing A and B must also contain the partition elements. In conclusion, the smallest field containing A and B coincides with that containing the partition {AB, A*Bᴅ, Aᴅ*B, Aᴅ*Bᴅ}.