Let S = {Barnsley, Manchester United, Southend, Sheffield United, Liverpool, Maroka Swallows, Witbank Aces, Royal Tigers} be a universal set, A = {Southend, Liverpool, Maroka Swallows, Royal Tigers}, and B = {Barnsley, Manchester United, Southend}. Find n((A B)').
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To find the cardinality of the set ((A ∩ B)'), we first need to understand the notation used.
Given sets A and B:
A ∩ B represents the intersection of sets A and B, which are the elements that are common to both sets.
(A ∩ B)' represents the complement of the intersection, which contains all the elements in the universal set S that are not in the intersection of A and B.
n(X) represents the cardinality (number of elements) of set X.
Now, let's find the cardinality of ((A ∩ B)') step by step:
1. Determine the intersection of sets A and B: {Southend}
- The intersection of A and B is the set that contains the elements that are present in both A and B.
2. Find the complement of the intersection:
- Since the universal set S contains all the elements, the complement of the intersection ((A ∩ B)') will be the set that contains all the elements in S that are not in the intersection {Southend}.
((A ∩ B)') = {Barnsley, Manchester United, Sheffield United, Maroka Swallows, Witbank Aces, Royal Tigers}
3. Calculate the cardinality of ((A ∩ B)'):
n((A ∩ B)') = Number of elements in the set ((A ∩ B)') = 6
Therefore, n((A ∩ B)') = 6.