Math_61_Final_2

22S-MATH-61-LEC-2 Final Exam CONNOR PEDERSEN TOTAL POINTS 39 / 40 QUESTION 1 1 Relative powerset 5 / 5 ✓ + 2 pts Correct answer for part 1 ($$8$$) + 1 pts Answer for part 1 is off by $$1$$ ✓ + 1 pts The number $$2$$ appears somewhere, and is raised to a power other than $$5$$ ✓ + 1 pts The expression $$n-k$$ appears somewhere ✓ + 1 pts Correct answer for part 2 ($$2^{n-k}$$) + 0 pts Incorrect or blank QUESTION 2 2 Euler/Hamilton Cycle 5 / 5 ✓ - 0 pts Correct - 1 pts No justification for (2) - 2 pts (2) incorrect - 5 pts Skipped QUESTION 3 3 Reflexive relations 4 / 5 ✓ + 1 pts The definition of reflexive is stated correctly OR at least one of the two directions ("if" and "only if") is proven correctly in part 1 + 1 pts Both directions ("if" and "only if") are proven correctly in part 1 ✓ + 1 pts $$|X\times X|=n^2$$ ✓ + 1 pts $$|\Delta|=n$$ ✓ + 1 pts Correct answer for part 3 ($$|\{R:\Delta\subseteq R\subseteq X\times X\}|$$, computed using part 2 and Problem 1 part 2) + 0 pts Blank or incorrect QUESTION 4 4 Recurrence relatoin 5 / 5 ✓ - 0 pts Correct - 0.5 pts Very minor issues with solution - 1 pts Minor issues with solution - 2 pts Issues with solution - 5 pts Blank/incorrect QUESTION 5 5 EXAMPLE 5 / 5 ✓ - 0 pts Correct - 1 pts Miscounted the 2E's case - 2 pts Completely omitted/didn't deal with the 2E's case - 2 pts Divided total by 2 and not just the 2E's case - 1 pts Miscounted the only 1 (or at most 1) E case - 3 pts Ignored the 4 letter part of the problem QUESTION 6 6 Cones and bones 5 / 5 ✓ + 2 pts Rearranging the equation into the equation of the type $$y_1+y_2+y_3=11$$ with $$y_i$$ all nonnegative and with only upper boundary conditions. Or equivalent progress ✓ + 2 pts Idea to count all solutions - bad solutions and recognizing that bad solutions can overlap. Or equivalent progress ✓ + 1 pts Correct applications of the cones and bones + 0 pts Not much progress - 1 pts Mistake - 2 pts Big mistake(s) QUESTION 7 7 Induction 5 / 5 ✓ + 1 pts Base case ✓ + 1 pts Clearly stated induction hypothesis ✓ + 3 pts Clear and correct Inductive step + 1.5 pts The inductive step is promising, but incomplete because of logical or other mistakes or

unclear reasoning - 1 pts Not enough explanation or slightly unclear reasoning or small mistakes QUESTION 8 8 Divisibility graph 5 / 5 ✓ - 0 pts Correct - 0.5 pts Minor issues with graph conceptualization - 0.5 pts Insufficient justification in parts 1/2 - 0.5 pts Minor issues in part 3 - 1 pts Issues in part 3 - 1 pts Minor issues with solution - 3 pts Major issues with solution - 4 pts Some good ideas, not a solution - 5 pts Blank/incorrect Page 2