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rec0524_key - COT 3100 Summer 2001 Recitation 05/24 Sets 1...

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COT 3100 Summer 2001 Recitation 05/24 Sets 1. Let U is all real numbers and consider the following sets, which are intervals of real numbers: A =[1,3)={ x | 1 x <3 } B = (2, 4]={ x | 2< x 4 } C = ( -∝ , 5) ={ x | x < 5) Express the following sets as intervals, or unions of disjoint intervals: a) ¬ C ¬ C = { x | x 5 } b) A - B A - B = { x |1 x 2 } c) A - C A - C= d) C - A C - A= { x | x < 1 or 3 x <5 } e) A B A B = { x | 1< x <3 } f) A B A B = { x | 1 x 4 } 2. Prove or disprove that for any sets A , B , C a) A B - A B =( A - B ) ( B - C ) A B - A B = { x | x A B x A B }..………by definition of the set difference = { x | x A B ¬ ( x A B )}..……………………by definition of = { x | ( x A x B ) ¬ ( x A x B )}..………by definition of and = { x | ( x A x B ) ( x A x B )}..………………by DeMorgan’s law = { x | [ x A ( x A x B )] [ x B ( x A x B )]}.… distributive law = { x | [( x A x A ) ( x A x B )] [( x B x A ) ( x B
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