Consider the following relation on set B = {a, b, {a}, {b}, {a, b}}: P = {(a, b), (b, {a, b}), ({a, b}, a), ({b}, a), (a, {a})}. Which one of the following sets is a partition S of B = {a, b, {a}, {b}, {a, b}}? 1. {{a, b, {a}, {b}}, {{a, b}}} 2. {{a}, {b}, {a, b}} 3. {{a, b, {a}}, {{a}, {b}, {a, b}}} 4. {a, b, {a}, {b}, {a, b}} (A partition of the given set B can be defined as a set S = {S1, S2, S3, …}. The members of S are subsets of B (each set Si is called a part of S) such that a. for all i, Si =/ 0/ (that is, each part is nonempty), b. for all i and j, if Si =/ Sj, then Si Sj = 0/ (that is, different parts have nothing in common), and c. S1 S2 S3 … = B (that is, every element in B is in some part Si). It is possible to form different partitions of B depending on which subsets of B are formed to be elements of S.
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Let us find out which one of the following sets is a partition of the set
1. For the family we have that
a.
b.
c.
Therefore, is partition of
2. For the family we have that , and hence this family is not a partition of
3. For the family we have that , and hence this family is not a partition of
4. For that set we have that , and hence this family is not a partition of