0, let P(n) be the formula + (2n – 1) + (2n +1)= (n + 1)2 . 1+3 5 _ Prove that P(n) is t” src=”https://files.transtutors.com/cdn/qimg/9d29f933b01a4dbf9a7797d8e9dc4c03.jpg” aria-describedby=”d62″>

(b) Is there a simple connected graph having 9 vertices and 7 edges? explain why Either draw such a graph no such graph exist

15. [10 роints] (а) Does a If you answer yes, draw at least one such a graph. If you answer no, explain why there is no such

8. [10 points] Give a (direct) proof of the following statement. Clearly describe your argument in full Claim. Let r be a rat

7. [10 points] Let N(S) denoted the number of elements of a set S. Consider the following information regarding three sets A,

18. [10 points] For each integer n > 0, let P(n) be the formula + (2n – 1) + (2n +1)= (n + 1)2 . 1+3 5 _ Prove that P(n) is true for every integer n 0 by mathematical induction (a) Write the basis step for the induction here: (b) Write the inductive step for the induction here: (b) Is there a simple connected graph having 9 vertices and 7 edges? explain why Either draw such a graph no such graph exists. or 15. [10 роints] (а) Does a If you answer yes, draw at least one such a graph. If you answer no, explain why there is no such a simple graph eight vertices of degrees 1, 1, 2, 2, 3, 3, 4, 4 exist? on graph We were unable to transcribe this image8. [10 points] Give a (direct) proof of the following statement. Clearly describe your argument in full Claim. Let r be a rational number and n > 0 be an integer. Then r” is a rational number. Proof: 7. [10 points] Let N(S) denoted the number of elements of a set S. Consider the following information regarding three sets A, B, and C all of which are subsets of a set U: N(A) = 14, N(B) List all statements which are true: 10, N(AUBUC) 24 and N(An B) 6 = = = (a) C has at most 24 elements (b) C has at least 6 elements (c) AUB has exactly 18 elements (d) BUC has exactly 15 elements

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