Prove that if r is a reflexive relation on a nonempty

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Prove that if R is a reflexive relation on a nonempty set A, then R R n for every positive integer n. // Remark: Evidently we need to do a proof by induction since the n th composition power of a relation is defined recursively. Proof. The basis step is true since R 1 = R implies R R 1 for any relation R on a nonempty set A. To deal with the induction step, it suffices to show that if the relation R is reflexive, then ( n ε + )( R R n R R n+1 ). To this end, pretend n is an arbitrary positive integer, R is reflexive, and R R n . Let (a,b) ε R be arbitrary. Then (a,b) ε R and the induction hypothesis, R R n , implies (a,b) ε R n . R reflexive and a ε A (a,a) ε R. The definition of the composition R n R = R n+1 , (a,a) ε R, and (a,b) ε R n implies (a,b) ε R n+1 . Thus we have shown that R R n R R n+1 for an arbitrary n ε + . So we have verified -- ugh -- that ( n ε + )( R R n R R n+1 ). Since we have verified the hypotheses of the induction axiom, modus ponens wraps things up: If R is reflexive, ( n ε + )( R R 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? V = E + 1 = 38 (b) What is the maximum number of leaves that a binary tree of height 6 can have? If l denotes the number of leaves, then l 2 6 = 64. The maximum occurs when the tree is complete, i.e., all the leaves are at the same level. (c) If a full 3-ary tree has 24 internal vertices, how many leaves does it have? A full 3-ary tree with 24 internal vertices must have V = (3) (24) + 1 = 73 vertices. Since 24 are internal, there must be l = 73 - 24 = 49 leaves. _________________________________________________________________ 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. Proof. We must establish that R is reflexive, symmetric, and transitive to show that it is an equivalence relation.
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