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1. Let \( A=\{1,2,3,4\} \) and \( B=\{a, c\} \). Write the following sets by listing their elements between braces: (a) \( A \times B \) (b) \( B \times B \) (c) \( \emptyset \times B \) (d) \( (A \times B) \times B \)

Ask by Hall Malone. in the United States
Mar 12,2025

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Answer

(a) \( A \times B = \{(1,a), (1,c), (2,a), (2,c), (3,a), (3,c), (4,a), (4,c)\} \) (b) \( B \times B = \{(a,a), (a,c), (c,a), (c,c)\} \) (c) \( \emptyset \times B = \emptyset \) (d) \( (A \times B) \times B = \{((1,a),a), ((1,a),c), ((1,c),a), ((1,c),c), ((2,a),a), ((2,a),c), ((2,c),a), ((2,c),c), ((3,a),a), ((3,a),c), ((3,c),a), ((3,c),c), ((4,a),a), ((4,a),c), ((4,c),a), ((4,c),c)\} \)

Solution

(a) To compute \( A \times B \), we pair each element of \( A \) with each element of \( B \): \[ A \times B=\{(1,a),\, (1,c),\, (2,a),\, (2,c),\, (3,a),\, (3,c),\, (4,a),\, (4,c)\} \] (b) For \( B \times B \), we pair each element of \( B \) with each element of \( B \): \[ B \times B=\{(a,a),\, (a,c),\, (c,a),\, (c,c)\} \] (c) The Cartesian product with the empty set always gives the empty set: \[ \emptyset \times B=\emptyset \] (d) For \( (A \times B) \times B \), we first form \( A \times B \) (as in part (a)) and then pair each element of \( A \times B \) with each element of \( B \): \[ \begin{aligned} (A \times B) \times B = \{ & ((1,a),a),\, ((1,a),c),\, ((1,c),a),\, ((1,c),c),\\[1mm] & ((2,a),a),\, ((2,a),c),\, ((2,c),a),\, ((2,c),c),\\[1mm] & ((3,a),a),\, ((3,a),c),\, ((3,c),a),\, ((3,c),c),\\[1mm] & ((4,a),a),\, ((4,a),c),\, ((4,c),a),\, ((4,c),c) \} \end{aligned} \]

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(a) The Cartesian product \( A \times B \) consists of all ordered pairs where the first element is from \( A \) and the second element is from \( B \). Therefore, \( A \times B = \{(1,a), (1,c), (2,a), (2,c), (3,a), (3,c), (4,a), (4,c)\} \). (b) The Cartesian product \( B \times B \) consists of all ordered pairs where both elements are from \( B \). Thus, \( B \times B = \{(a,a), (a,c), (c,a), (c,c)\} \). (c) The Cartesian product \( \emptyset \times B \) is the set of all ordered pairs where the first element is from the empty set \( \emptyset \) and the second is from \( B \). Since there are no elements to pair with, \( \emptyset \times B = \emptyset \). (d) The Cartesian product \( (A \times B) \times B \) consists of all ordered pairs where the first element is from \( A \times B \) and the second is from \( B \). Therefore, \( (A \times B) \times B = \{((1,a),a), ((1,a),c), ((1,c),a), ((1,c),c), ((2,a),a), ((2,a),c), ((2,c),a), ((2,c),c), ((3,a),a), ((3,a),c), ((3,c),a), ((3,c),c), ((4,a),a), ((4,a),c), ((4,c),a), ((4,c),c)\} \).

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