3 I need help answering the following set theory question pl

3. I need help answering the following set theory question, please. Thanks a lot.

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Solution

3(a) X = {a, b, c}

|X| = 3

P(X) = Power set of X = {S: S is a subset of X}

|P(X)| = 2³ = 8

A hypergraph is a family of sets drawn from X.

Different hypergraphs:

{, X}, {, X, {a}}, {, X, {b}}, {, X, {c}}, {, X, {a, b}}, {, X, {a, c}}, {, X, {b, c}}, {, X, {a}, {a, b}}, {, X, {b}, {a, b}}, {, X, {a}, {a, c}}, {, X, {c}, {a, c}}, {, X, {b}, {b, c}}, {, X, {c}, {b, c}}, {, X, {a}, {b}, {a, b}}, {, X, {a}, {c}, {a, c}}, {, X, {b}, {c}, {b, c}}, P(X), {, X, {a}, {b, c}}, {, X, {b}, {a, c}}, {, X, {c}, {a, b}}

{, X, {a}, {b}}, {, X, {a}, {c}}, {, X, {b}, {c}}, {, X, {a}, {b}, {c}}, {, X, {a}, {b}, {c}, {a, b}}, {, X, {a}, {b}, {c}, {b, c}}, {, X, {a}, {b}, {c}, {a, c}}, {, X, {a}, {b}, {b, c}}, {, X, {a}, {b}, {a, c}}, {, X, {a}, {c}, {b, c}}, {, X, {a}, {c}, {a, b}}, {, X, {b}, {c}, {a, b}}, {, X, {b}, {c}, {a, c}}

denotes null set.

(b) From the collection of hypergraphs, the topologies are:

{, X}, {, X, {a}}, {, X, {b}}, {, X, {c}}, {, X, {a, b}}, {, X, {a, c}}, {, X, {b, c}}, {, X, {a}, {a, b}}, {, X, {b}, {a, b}}, {, X, {a}, {a, c}}, {, X, {c}, {a, c}}, {, X, {b}, {b, c}}, {, X, {c}, {b, c}}, {, X, {a}, {b}, {a, b}}, {, X, {a}, {c}, {a, c}}, {, X, {b}, {c}, {b, c}}, P(X), {, X, {a}, {b, c}}, {, X, {b}, {a, c}}, {, X, {c}, {a, b}}

(c) T₁ and T₂ be two topologies on X given by

T₁ = {, X, {a}, {a, b}}, T₂ = {, X, {a}, {b, c}}

The smallest topology on X containing T₁ and T₂ is T₁T₂ = {, X, {a}}

(d) A and B be two topologies on X given by

A = {, X, {a}, {a, b}}, B = {, X, {c}, {a, c}}

These topologies are homeomorphic because both the topologies are finite and having the same cardinality, i.e. |A| = |B| = 4

There exists a bijection as , X X, {a} {c}, {a, b} {a, c}