# Also recall that the n th composition power of a

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Also, recall that the n th composition power of a relation on a set is defined recursively by R 1 = R, and for each n ε , if n 1, then R n+1 = R n R. Prove that if R is a reflexive relation on a nonempty set A, then R R n for every positive integer n.

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TEST3/MAD2104 Page 4 of 4 _________________________________________________________________ 7. (15 pts.) (a) How many vertices does a tree with 37 edges have? (b) What is the maximum number of leaves that a binary tree of height 6 can have? (c) If a full 3-ary tree has 24 internal vertices, how many leaves does it have? _________________________________________________________________ 8. (10 pts.) Suppose that A is the set consisting of all real- valued functions with domain consisting of the interval [-1,1] . Let R be the relation on the set A defined as follows: R = { (f,g) ( C)( C ε and f(0) - g(0) = C) } Prove that R is an equivalence relation on the set A.
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