Answer
a) A ∪ B = (107 ⋅ personal computers, 44 ⋅ routers, 6 ⋅ servers, 2 ⋅ main frames)
b) A ∩ B = (14 ⋅ personal computers, 6 ⋅ routers)
c) A - B = (2 ⋅ main frames)
d) A + B = (121 ⋅ personal computers, 50 ⋅ routers, 6 ⋅ servers, 2 ⋅ main frames)
Work Step by Step
A) Department A requires a specific set of equipment, and Department B requires another set. When considering both departments using the same equipment, the combined set is represented as A∪B, which contains 107 personal computers, 44 routers, 6 servers, and 2 main frames. In the union, the multiplicity of an element is determined by the highest multiplicity of that element in either set.
B)Department A requires a specific set of equipment, and Department B requires another set. The shared equipment between the two departments consists of personal computers and routers, with the quantities specified as 14 and 6, respectively. The intersection of the two sets is determined based on the lowest multiplicity of each element in either set. If there are no common elements between the sets, they won't be part of the intersection.
C) The equipment required by Department A is distinct from the equipment required by Department B. The equipment used by Department B but not by Department A is represented by the difference of the sets, denoted as B-A.
The multiplicity of an element in the difference is calculated by subtracting the multiplicity in B from the multiplicity in A. If the result is negative, it means that the element is not present in the set.
For instance, the difference B-A consists of 2 main frames, which means that Department B uses 2 more main frames than Department A. Other equipment like personal computers, routers, and servers are not used by Department B but are used by Department A, resulting in negative multiplicities, indicating that they are not present in B-A.
B-A = {(−93)⋅personal computers, (−38)⋅routers, (−6)⋅servers, 2⋅main frames}.
When the multiplicity of an element in the difference set is negative, it indicates that the element is not part of the set.
B−A={2⋅main frames}
D) Department A requires a specific set of equipment, while Department B requires another set. The equipment used by both departments is represented by the union of the intersection of the two sets (shared equipment) with itself and the differences (equipment used by one department but not the other).
In mathematical notation, A+B = (A∩B)∪(A∩B)∪(A−B)∪(B−A).
The multiplicity of an element in the difference set is calculated by subtracting the multiplicity in B from the multiplicity in A. If the result is negative, it indicates that the element is not present in the set.
For instance, A-B = {93⋅personal computers, 38⋅routers, 6⋅servers}, meaning Department A requires 93 more personal computers, 38 more routers, and 6 more servers than Department B.
The multiplicity of an element in the union set is determined by taking the highest multiplicity of the element from either set.
A+B = {(14+14+93)⋅personal computers, (6+6+38)⋅routers, 6⋅servers, 2⋅main frames}, which means that the union set includes 121 personal computers, 50 routers, 6 servers, and 2 main frames, considering the highest multiplicities from both departments.
