A b c d 6 10 pts provide a proof by induction that 2

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a b \ \ \ c d 6. (10 pts.) Provide a proof by induction that 2 n 2n for every positive integer n. Be explicit concerning your use of the induction hypothesis in the inductions step.
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TEST-3/MAD2104 Page 4 of 4 7. (15 pts.) (a) How many edges does a tree with 37 vertices have? (b) What is the maximum number of leaves that a binary tree of height 10 can have? (c) If a full 3-ary tree has 24 internal vertices, how many vertices does it have? 8. (10 pts.) Suppose that R is an equivalence relation on a nonempty set A. Recall that for each a ε A, the equivalence class of a is the set [a] = {s | (a,s) ε R}. Prove the following proposition: If (a,b) ε R, then [a] = [b]. Hint: The issue is the set equality, [a] = [b], under the hypothesis that (a,b) ε R. So pretend (a,b) ε R and use this to show s ε [a] s ε [b], and s ε [b] s ε [a]. Be explicit regarding your use of the relational properties of R.
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