Cartesian product denition the cartesian products of

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Unformatted text preview: | = 3 4.  |{ø}| = 1 5.  The set of integers is infinite. Power Sets Definition: the set of all subsets of a set A, denoted P(A), is called the power set of A. Example: if A = {a, b} then P(A) = { ø, {a}, {b}, {a, b} } —༉  If a set A has n elements, then the cardinality of its power set is 2ⁿ. —༉  In previous example, n=2 and |P(A) |= 22 = 4 Tuples —༉  The ordered n- tuple ( a1, a2, …, an ) is the ordered collection that has a1 as its first element, a2 as its second element and so on until an as its last element. —༉  Two n- tuples are equal if and only if their corresponding elements are equal. —༉  2- tuples are called ordered pairs. —༉  The ordered pairs (a , b ) and (c , d ) are equal if and only if a = c and b = d. Cartesian Product Definition: the Cartesian product of two sets A and B, denoted by A × B is the set of ordered pairs (a, b) where a ∈ A and b ∈ B. Example: A = { a , b } B = { 1 , 2 , 3 } A × B = { ( a , 1 ) , ( a , 2 ) , ( a , 3 ) , ( b , 1 ) , ( b , 2 ) , ( b , 3 ) } Definition: a subset R of the Cartesian product A × B is called a relation from the set A to the set B. Cartesian Product Definition: the Cartesian products of the sets A1, A2, …, An denoted by A1 × A2 × … × An , is the set of ordered n- tuples (a1, a2, …, an) where ai belongs to Ai for i = 1, …, n. Example: what is A × B × C where A = { 0 , 1 }, B = { 2 , 3 } and C = { 4 , 5 }. Solution: A × B × C = { ( 0 , 2 , 4 ) , ( 0 , 2 , 5 ) , ( 0 , 3 , 4 ) , ( 0 , 3 , 5 ) , ( 1 , 2 , 4 ) , ( 1 , 2 , 5 ) , ( 1 , 3 , 4 ) , ( 1 , 3 , 5 ) } Truth Sets of Quan$fiers —༉  Given a predicate P(x) and a domain D, we define the truth set of P(x) to be the set of elements...
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