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171sp05_midterm2

# 171sp05_midterm2 - 1 Â 10 r n Â 2 Prove if n 2 N r n = 2 n...

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550.171 Discrete Mathematics - Spring 2005 Midterm #2 Name 1. [15 pt] Prove: if G is a connected graph with n vertices and n edges ( n ¸ 3), then G contains a cycle. 2. Let A and B be sets. (a) [3 pts] Brie°y explain what is meant by f : A ! B . (b) [7 pts] Suppose f : A ! B . For each b 2 B , let A b = f a 2 A : f ( a ) = b g , that is, A b is the set of points in the domain of f that f sends to b in its image. Provide de¯nitions of what it means for f to be one-to-one , and for f to be onto in terms of the sets A b .

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3. Please provide a short answer to each of the following. (a) [8 pts] How many distinct 2-colorings of K 5 (the complete graph on 5 vertices) are there? (b) [3 pts] ( continuation ) How many distinct 2-colorings of K 5 use both colors? (c) [9 pts] Find a set A and a function f : A ! A such that f is one-to-one (so that f ¡ 1 exists), but dom( f ¡ 1 ) 6 = A . 4. [10 pt] Prove: If A and B are sets such that A \ B = ; , then ( A £ B ) \ ( B £ A ) = ; . Reminder: the notation C £ D is the set of all ordered pairs ( c;d ) with c 2 C and d 2 D .
5. [20 pts] Let r 0 = 2, r 1 = 7, and for integer n ¸ 2, r n = 7 r n ¡

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Unformatted text preview: 1 Â¡ 10 r n Â¡ 2 . Prove: if n 2 N , r n = 2 n +5 n . 6. Consider the graph G pictured to the right. (a) [3 pt] Find the chromatic number of G , Ã‚ ( G ). (b) [5 pt] Identify (i) an independent set that is not maximal, (ii) a maximal independent set; and (iii) compute the independence number of G , Â® ( G ). Clearly label each. (c) [2 pt] Yes or No: Are there any maximal independent sets in G that are not maximum? 7. TRUE or FALSE . Spell out the word you choose. (a) A connected graph must have a spanning tree. (b) Every tree with two or more vertices must have at least two leaves. (c) If f : A ! B is one-to-one, and dom( f Â¡ 1 ) = B , then f is a bijection. (d) If the average degree of a connected graph is two, then the graph must be a tree. (e) There are 2 Â³ n 2 Â´ induced subgraphs of K n ....
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171sp05_midterm2 - 1 Â 10 r n Â 2 Prove if n 2 N r n = 2 n...

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