Question

Suppose \(A=\\{b, c, d\\}\) and \(B=\\{a, b\\}\). Find: (a) \((A \times B) \cap(B \times B)\) (b) \((A \times B) \cup(B \times B)\) (c) \((A \times B)-(B \times B)\) (d) \((A \cap B) \times A\) (e) \((A \times B) \cap B\) (f) \(\mathscr{P}(A) \cap \mathscr{P}(B)$$(\mathbf{g}) \mathscr{P}(A)-\mathscr{P}(B)\) (h) \(\mathscr{P}(A \cap B)\) (i) \(\mathscr{P}(A) \times \mathscr{P}(B)\)

Short answer

(a) \((A \times B) \cap (B \times B) = \{ (b,a), (b,b)\} (b) \((A \times B) \cup (B \times B) = \{ (b,a), (b,b), (c,a), (c,b), (d,a), (d,b), (a,a), (a,b), (b,a) \} (c) \((A \times B) - (B \times B) = \{ (c,a), (c,b), (d,a), (d,b) \} (d) \((A \cap B) \times A = \{(b,b),(b,c),(b,d)\} (e) \((A \times B) \cap B = \emptyset (f) \(\mathscr{P}(A) \cap \mathscr{P}(B) = \{ \emptyset, \{b\} \} (g) \(\mathscr{P}(A)-\mathscr{P}(B) = \{ \{c\}, \{d\}, \{b,c\}, \{b,d\}, \{c,d\}, \{b,c,d\} \} (h) \(\mathscr{P}(A \cap B) = \{ \emptyset, \{b\} \} (i) \(\mathscr{P}(A) \times \mathscr{P}(B) = \{ \(\emptyset,a\), \(\emptyset,b\),\((b,a)\), \((b,b)\), \((c,c)\), \(\{d,c\}\), \(\{b,c\},b\), \(\{b,c\},a\), \(\{b,c\},d\), \((\{b,c,d\},a)\), (\{b,c,d\},b\), (\{b,c,d\},c\), (\{b,c,d\},d\), (\{b,c,d\},(\{b,a\}\)) \}\)

Step-by-step solution

- Compute Cartesian Products

Firstly, compute the Cartesian Products \(A \times B\) and \(B \times B\). \n The Cartesian product of two sets is the set of all ordered pairs that can be formed, where the first element of each pair comes from the first set, and the second element comes from the second set. \(A \times B = \{ (b,a), (b,b), (c,a), (c,b), (d,a), (d,b) \}\) and \(B \times B = \{ (a,a), (a,b), (b,a), (b,b)\}\).

- Find Intersections, Unions and Differences

Next, compute the Intersection, Union and Difference of these Cartesian products. \n The Intersection of two sets is the set of elements which are in both sets. The Union of two sets is the set of elements which are in either set. The difference of two sets is the set of elements which are in the first set but not in the second set. Hence, (a) \((A \times B) \cap (B \times B) = \{ (b,a), (b,b)\}\), (b) \((A \times B) \cup (B \times B) = \{ (b,a), (b,b), (c,a), (c,b), (d,a), (d,b), (a,a), (a,b), (b,a) \}\), and (c) \((A \times B) - (B \times B) = \{ (c,a), (c,b), (d,a), (d,b) \}\).

- Compute Intersection of Sets and Cartesian Product

Now, find the intersection of A and B, and then compute the Cartesian product with A. \n The intersection of A and B would be \(A \cap B = \{b\}\). Hence, (d) \((A \cap B) \times A = \{(b,b),(b,c),(b,d)\}\).

- Intersection of Cartesian Product with Set B

Next, compute the Intersection of the Cartesian product of A and B with the set B. \n As B does not contain ordered pairs, this intersection will be an empty set, represented as \(\emptyset\). Hence, (e) \((A \times B) \cap B = \emptyset\).

- Compute Power Sets and their Interactions

Finally, compute the power set of sets, which is the set of all subsets of a given set, including the set itself and the empty set. Here, Power Set of A would be \( \mathscr{P}(A) = \{ \emptyset, \{b\}, \{c\}, \{d\}, \{b,c\}, \{b,d\}, \{c,d\}, \{b,c,d\} \}\) and Power Set of B would be \( \mathscr{P}(B) = \{ \emptyset, \{a\}, \{b\}, \{a,b\} \}\). Hence, (f) \(\mathscr{P}(A) \cap \mathscr{P}(B) = \{ \emptyset, \{b\} \}\), (g) \(\mathscr{P}(A)-\mathscr{P}(B) = \{ \{c\}, \{d\}, \{b,c\}, \{b,d\}, \{c,d\}, \{b,c,d\} \}\), (h) \(\mathscr{P}(A \cap B) = \{ \emptyset, \{b\} \}\), (i) \(\mathscr{P}(A) \times \mathscr{P}(B) = \{ \(\emptyset,a\), \(\emptyset,b\),\((b,a)\), \((b,b)\), \((c,c)\), \(\{d,c\}\), \(\{b,c\},b\), \(\{b,c\},a\), \(\{b,c\},d\), \((\{b,c,d\},a)\), (\{b,c,d\},b\), (\{b,c,d\},c\), (\{b,c,d\},d\), (\{b,c,d\},(\{b,a\}\)) \}\).