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.
Let us find out which one of the following sets is a partition "S" of the set "B = \\{a, b,\\{a\\}, \\{b\\}, \\{a, b\\}\\}."
1. For the family "S=\\{\\{a, b, \\{a\\}, \\{b\\}\\}, \\{\\{a, b\\}\\}\\}" we have that
a. "\\{a, b, \\{a\\}, \\{b\\}\\}\\ne\\emptyset,\\ \\{\\{a, b\\}\\}\\ne\\emptyset"
b. "\\{a, b, \\{a\\}, \\{b\\}\\}\\cap \\{\\{a, b\\}\\}=\\emptyset"
c. "\\{a, b, \\{a\\}, \\{b\\}\\}\\cup\\{\\{a, b\\}\\}=B"
Therefore, "S" is partition of "B."
2. For the family "\\{\\{a\\}, \\{b\\}, \\{a, b\\}\\}" we have that "\\{b\\}\\cap\\{a, b\\}=\\{b\\}\\ne\\emptyset", and hence this family is not a partition of "B."
3. For the family "\\{\\{a, b, \\{a\\}\\}, \\{\\{a\\}, \\{b\\}, \\{a, b\\}\\}\\}" we have that "\\{a, b, \\{a\\}\\}\\cap\\{\\{a\\}, \\{b\\}, \\{a, b\\}\\}=\\{\\{a\\}\\}\\ne\\emptyset", and hence this family is not a partition of "B."
4. For that set "\\{a, b, \\{a\\}, \\{b\\}, \\{a, b\\}\\}" we have that "\\{a\\}\\cap \\{a, b\\}=\\{a\\}\\ne\\emptyset", and hence this family is not a partition of "B."
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