04 - Fall 2009 CS131 – Combinatorial Structures Homework...

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Unformatted text preview: Fall 2009 CS131 – Combinatorial Structures Homework 4 Homework 4, due Oct 15 You must prove your answer to every question. Do not rely only on the homework for exercise: there are several self-check ex- ercises of the easier kind in the book, try to solve them, too! Problem 1. (10pt) What is the number of surjective functions from {1,2,..., n } to {1,2,3}? [Hint: recall the similar simpler problem from the last homework where the range had size 2.] Solution. We must subtract from this the number of functions with a range size exactly 2 (2 n- 2 functions for range {1,2}, similarly for ranges {1,3} and {2,3}), and also the number of functions with range size 1 (there are just 3 functions, with ranges {1}, {2} and {3}). The result is 3 n- 3(2 n- 2)- 3 = 3 n- 3 · 2 n + 3. Problem 2. (10pt) How many different symmetric, reflexive relations are there in the set {1,2,..., n }? Solution. All pairs ( x , x ) are in the relation, we do not have a choice. For each unordered pair { x , y } we can decide independently from the others whether to take ( x , y ) and ( y , x ) into the relation (if we take one, we must also take the other). There are ( n 2 ) unordered pairs, so the number of possibilities is 2 ( n 2 ) . Problem 3 (Lipschutz-Lipson) . Each of the following defines a relation on the pos- itive integers: (1) “ x is greater than y .” (3) x + y = 10....
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04 - Fall 2009 CS131 – Combinatorial Structures Homework...

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