College Algebra Exam Review 53

College Algebra Exam Review 53 - 63 1.9 COUNTING the...

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Unformatted text preview: 63 1.9. COUNTING the intersection of two sets is the product of the characteristic functions: 1X \Y D 1X 1Y . Let X 0 denote the relative complement of a subset X Â U ; that is, 0 D U n X. X Proposition 1.9.11. Let A1 ; A2 ; : : : ; An be subsets of U . Then (a) X X X 1Ai C 1Ai \Aj 1Ai \Aj \Ak 1A01 \A02 \ \A0n D 1 i i <j n C i <j <k C . 1/ 1A1 \ \An : (b) 1A1 [A2 [ [An D X 1Ai i X X 1Ai \Aj C i <j C . 1/n 1Ai \Aj \Ak i <j <k 1 1A1 \ An : Proof. For part (a), 1A01 \A02 \ \A0 n D 1A01 1A02 D .1 D1 1A0n 1A1 /.1 1A2 / .1 1An / X X X 1Ai C 1Ai 1Aj 1Ai 1Aj 1Ak i C . 1/ 1A1 1A2 X 1Ai C 1Ai \Aj 1An X i D1 i <j <k C X i <j n i <j n i <j <k C C . 1/ 1A1 \ 1Ai \Aj \Ak \An : Part (b) follows from part (a), because A0 \ A0 \ 1 2 \ A0 D .A1 [ A2 [ n [ An /0 : I Corollary 1.9.12. Suppose that U is a finite set and that A1 ; A2 ; : : : ; An are subsets of U . Then ...
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This note was uploaded on 12/15/2011 for the course MAC 1105 taught by Professor Everage during the Fall '08 term at FSU.

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