Unformatted text preview: . ( 30-15+2 2 ) ( 2 ) Given two sets A and B , let m = | A | , n = | B | . Type Deﬁnition Number correspondence no restrictions (2 n ) m = 2 mn function ∀ a ∈ A, ∃ unique b ∈ B, s.t., f ( a ) = b n m surjection ( onto ) function + ∀ b ∈ B, ∃ a ∈ A, s.t., f ( a ) = b n ! S ( m, n ) injection ( one-to-one ) function + ∀ x 1 , x 2 ∈ A, x 1 6 = x 2 → f ( x 1 ) 6 = f ( x 2 ) n ( n-1) ··· ( n-m + 1) = n ! ( n-m )! ( 3 ) Cardinality. Two sets have the same cardinality if there is a bijection between them. (a) Does a line segment of 2 inches contain more points than a line segment of 1 inch? (b) Are there more integers than even integers? (c) Is there a bijection between natural numbers and real numbers? (d) For any set A (ﬁnite or inﬁnite), is there a bijection between A and P ( A ), the power set of A ? 1...
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- Spring '07
- Algorithms, Natural number, integer solutions