Let U = {u, n, i, t, e}
A = {n, i, t}
B = {n, e}
C = {u, n, i, t, e}
D = {u, e}
Find each of the following:
(a) A ¡ä B
(b) C ¡å D
(c) ¡D¡
(d) ¡A ¡å D¡
(e) B ¡ä ¡C¡
(f) (B ¡ä C) ¡ä D
NOTE: ¡å denotes ¡§union¡¨ , ¡ä denotes ¡§intersection¡¨ and ¡ ¡ denotes ¡§negation¡¨
To find the answers to these questions, we need to understand the operations of union (¡å), intersection (¡ä), and negation (¡).
(a) A ¡ä B:
The intersection of A and B is the set of elements that are common to both A and B. In this case, A = {n, i, t} and B = {n, e}. The common element between the two sets is "n", so A ¡ä B = {n}.
(b) C ¡å D:
The union of C and D is the set of elements that are in either set C or set D. In this case, C = {u, n, i, t, e} and D = {u, e}. The elements in both sets combined are {u, n, i, t, e}, so C ¡å D = {u, n, i, t, e}.
(c) ¡D¡:
Negation is the operation of finding the complement of a set. In this case, D = {u, e}. The complement of D, denoted as ¡D¡, is the set of all elements that are not in D. Therefore, ¡D¡ = {n, i, t}.
(d) ¡A ¡å D¡:
This expression involves the intersection of the negation of A and the complement of D. First, we find the complement of D: ¡D¡ = {n, i, t}. Then, we find the negation of A, which is the set of all elements that are not in A. Since U = {u, n, i, t, e}, and A = {n, i, t}, the elements not in A are {u, e}. Finally, we find the intersection of these two sets: {u, e} ¡ä {n, i, t} = {}. Therefore, ¡A ¡å D¡ = {}. (An empty set)
(e) B ¡ä ¡C¡:
Similar to the previous questions, we first find the complement of C: ¡C¡ = {}. Then, we find the intersection of B and the complement of C: B ¡ä {}. Since {} is empty and there are no common elements between B and ¡C¡, the intersection is also empty. Therefore, B ¡ä ¡C¡ = {}.
(f) (B ¡ä C) ¡ä D:
First, we find the intersection of B and C: B ¡ä C = {n}. Then, we find the intersection of this result and D: {n} ¡ä D = {}. Therefore, (B ¡ä C) ¡ä D = {}. (An empty set)