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

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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) $$ A 	imes B $$ (b) $$ B 	imes B $$ (c) $$ \emptyset 	imes B $$ (d) $$ (A 	imes B) 	imes B $$

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**(a)**  
$$ A 	imes B = \{ (1,a),\; (1,c),\; (2,a),\; (2,c),\; (3,a),\; (3,c),\; (4,a),\; (4,c) \} $$

**(b)**  
$$ B 	imes B = \{ (a,a),\; (a,c),\; (c,a),\; (c,c) \} $$

**(c)**  
$$ \emptyset 	imes B = \emptyset $$  
*(Because the Cartesian product with the empty set is empty.)*

**(d)**  
$$ (A 	imes B) 	imes B = \{ ((1,a),a),\; ((1,a),c),\; ((1,c),a),\; ((1,c),c),\; ((2,a),a),\; ((2,a),c),\; ((2,c),a),\; ((2,c),c),\; ((3,a),a),\; ((3,a),c),\; ((3,c),a),\; ((3,c),c),\; ((4,a),a),\; ((4,a),c),\; ((4,c),a),\; ((4,c),c) \} $$
**(a)** $$ A 	imes B = \{ (1,a),\; (1,c),\; (2,a),\; (2,c),\; (3,a),\; (3,c),\; (4,a),\; (4,c) \} $$ **(b)** $$ B 	imes B = \{ (a,a),\; (a,c),\; (c,a),\; (c,c) \} $$ **(c)** $$ \emptyset 	imes B = \emptyset $$ **(d)** $$ (A 	imes B) 	imes B = \{ ((1,a),a),\; ((1,a),c),\; ((1,c),a),\; ((1,c),c),\; ((2,a),a),\; ((2,a),c),\; ((2,c),a),\; ((2,c),c),\; ((3,a),a),\; ((3,a),c),\; ((3,c),a),\; ((3,c),c),\; ((4,a),a),\; ((4,a),c),\; ((4,c),a),\; ((4,c),c) \} $$

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) \( (A \times B) \times B \)

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