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# HW1 - Mathematics Department UCLA T Richthammer winter 09...

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Mathematics Department, UCLA T. Richthammer winter 09, sheet 1 Jan 02, 2009 Homework assignments: Math 170A Probability, Sec. 1 001. Let E 1 , E 2 , . . . be subsets of a universal set S . Draw Venn diagrams for the following sets: (a) E 1 E 2 E 3 , (b) E 1 ( E 2 - E 3 ) (c) E c 2 ( E 1 E c 3 ) (d) E 1 S 5 i =2 E i , where E 2 , . . . E 5 are disjoint Answer: pictures . .. 002. Prove ( E 1 E 2 ) c = E c 1 E c 2 and ( E 1 E 2 ) c = E c 1 E c 2 (special cases of de Morgan’s laws): (a) by drawing Venn-diagrams (b) by showing that x LHS x RHS . Answer: (a) pictures . .. (b) x ( E 1 E 2 ) c not ( x E 1 E 2 ) not ( x E 1 and x E 2 ) (not x E 1 ) or (not x E 2 ) x E c 1 or x E c 2 x E c 1 E c 2 . x ( E 1 E 2 ) c not ( x E 1 E 2 ) not ( x E 1 or x E 2 ) (not x E 1 ) and (not x E 2 ) x E c 1 and x E c 2 x E c 1 E c 2 . 003. Prove the distributive law ( E 1 E 2 ) F = ( E 1 F ) ( E 2 F ) (a) by drawing Venn-diagrams (b) by showing that x LHS ⇐⇒ x RHS . Answer:

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HW1 - Mathematics Department UCLA T Richthammer winter 09...

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