Final answer:
Set B, being a subset of A and having intersection AnB = {1,2,3} with set A, and given that the union AuB = {1,2,3,4,5,6}, implies that set B must be {1,2,3}. The union tells us which elements are in at least one of the sets, and the intersection tells us which are in both. Therefore, B contains only elements found in both A and B.
Step-by-step explanation:
To find set B given that BcA, AnB = {1,2,3} and AuB = {1,2,3,4,5,6}, we first need to understand the notation. BcA means that set B is a subset of set A, AnB represents the intersection of sets A and B, and AuB represents the union of sets A and B.
Since the union of sets A and B (AuB) includes all elements that are in A, in B, or in both, and we have AuB = {1,2,3,4,5,6}, we know that sets A and B together contain these numbers. The intersection of A and B (AnB), are elements that are common to both A and B, which is {1,2,3}.
As B is a subset of A (BcA), all elements of B must also be in A. Therefore, we can deduce that the elements of B must include {1,2,3} and could include additional elements from set A that are not listed in AnB. However, since AuB contains only {1,2,3,4,5,6}, and we've accounted for {1,2,3} in the intersection, the remaining elements {4,5,6} must be the elements that are in A but not in B (as B is a subset of A).
Thus, with the given information, we can conclude that set B is precisely the same as the intersection set, which is B = {1,2,3}.