# Cartesian product denition the cartesian products of

This preview shows page 1. Sign up to view the full content.

This is the end of the preview. Sign up to access the rest of the document.

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...
View Full Document

{[ snackBarMessage ]}

Ask a homework question - tutors are online