lab 1 and key

quiz 1 and key

Grading rubric:

• Page 1 (50%)
  • attempted all problems
  • have term definitions
  • all definitions are propositions
  • all definitions are atomic propositions
  • all definitions are from text
  • no part of text left out
  • have formula
  • 1st formula correct
  • 2nd formula correct
  • 3rd formula correct
• Page 2 (50%)
  • attempted all problems
  • one when a ∨ b
  • three contains negation of their one logic (¬(a ∨ b) unless errors with one)
  • three is equivalent to ¬a ∧ ¬b ∧ ¬c
  • reasonable logic syntax
  • A ⊕ C column is 01011010
  • B ↔︎ C column is 10011001
  • (2 points) center column ↔︎ of other two (00111100 unless errors above)

lab 2 and key

quiz 2 and key

Grading rubric:

• Page 1 (25%)
  • start with (P ∧ ¬Q)
  • logic syntax used
  • attempted a full proof
  • applied rules correctly
  • no skipped steps
  • end with ¬(P → Q)
• Page 2 (75%)
  • used same variable in all three blanks
  • wrote something in all four areas
  • got both case 1 expressions to same form
  • … with the case assumption correctly inserted
  • … using valid equivalence rules
  • … expressed in prose
  • got both case 2 expressions to same form
  • … with the case assumption correctly inserted
  • … using valid equivalence rules
  • … expressed in prose

lab 3 and key

quiz 3 and key

Grading rubric:

• Page 1 (40%)
  • no G are F
  • everything is F
  • nothing is G
  • uses therefore symbol
  • all G are F
  • something is G
  • some G is F
  • in the right order with no extras
• Page 2 (60%)
  • first: uses M and Z
  • first: universal or not-exist quantifier
  • first: logically correct
  • second: uses L and b
  • second: universal quantifier
  • second: implication
  • second: L(x,b) → L(b,x) not the other way around
  • third: universal or not-exist
  • third: allows both artist and champion to love
  • third: …only if they share no love

lab 4 and key

quiz 4 and key

Grading rubric:

• B has 1,4,9 (half credit for 1,2,3)
• B has 0 and no extra elements
• C has {} (half credit if C is {})
• C has {4}, {9}, {4,9} and no extra elements
• (2 points) A ∪ B has all of {0,2,3} and all of B ({0,1,2,3,4,9} unless B wrong)
• A ∪ B has nothing else, with no element listed twice
• (2 points) A ∩ B has only elements A has, and only elements B has ({0} unless B wrong)
• A ∩ B has all such elements
• A ∖ B has only elements A has, and no elements B has ({2,3} unless B wrong)
• A ∖ B has all such elements
• B ∪ C has both numbers and sets
• ⊕-set is correct ({1, 2, 3, 4, 9} unless B is wrong)
• ∀-set is B ∖ A ({1, 4, 9} unless B is wrong)
• ∃-set is B ∩ {4, 9}

lab 5 and key

quiz 5 and key

Grading rubric:

1. {(4,1), (4,2), (1,1), (1,2)}
2. {(4,1,3,3,3), (4,2,3,3,3)} – extra parens like ((4,1), (3,3,3)) OK
3. {(∅,∅)}
4. two of aok, oka, and aaa
5. MTHMTCS
6. {0, 1, 4} – half credit if has 1 twice
7. is defined as natural for some natural numbers but not all
8. is not invertible with the domain and co-domain of ℕ
9. b = 3a or equivalent – half-credit for a = 3b
10. has at least one element of domain related to 2+ elements in co-domain

lab 6 and key

quiz 6 and key

Grading rubric:

• Page 1 (50%)
  • base case includes 0
  • reasonable defense of base case being finite
  • induce on symbol (e.g. n), not specific number
  • next case is +1 (e.g., n+1)
  • appeals to addition of finite being finite
• Page 2 (50%)
  • definition of x is mathematical, larger, and natural
  • defense of x being natural fits definition of x
  • defense of x being natural fits definition of natural
  • last blank mentions assuming led to contradiction
  • nothing else wrong with proof

lab 7 and key

quiz 7 and key

Grading rubric:

1. 52 choose 5 = 2598960 (half for 52! / (52-5)! or 52⁵)
2. 8! = 40320
3. 8! / 2!3!2! = 1680
4. 7776⁶
5. 7776! / 7770! (half for 7776 choose 6)
6. 15
7. 3/64
8. 40/57
9. 40/60 = 2/3
10. 1/1000 (half for 1/500; allow (999/1000)⁵⁰⁰ in front at no penalty)

lab 8 and key

quiz 8 and key

There is no rubric because the quiz version printed and shown in class contained an error large enough that the quiz was dropped entirely.

lab 9 and key

quiz 9 and key

Grading rubric:

• has a base case
• base case includes -1
• base case shows both sides equal
• has inductive step
• inductive case assumes true at variable
• inductive case shows true at variable + 1
• induction argument uses algebra
• algebra correct
• has conclusion
• structure: introduces induction, labels parts, etc

lab 10 and key

quiz 10 and key

Grading rubric:

• Q1 is 2 · 2 · 3 · 5 (or 2² · 3 · 5)
• Q2 is 3^y = x
• Q3 is log_c(b) ÷ log_c(a) – half credit if inverse of that
• Q4 is 2 lg(a) + lg(b)
• Q5 is 3/2
• pf: each step follows from one above
• pf: ends with only integers and powers on last line
• pf: fits the rest of proof (e.g. 3^b = 2^a)

lab 11 and key

quiz 11 and key

Grading rubric:

• makes assumption
• assumption is negation of theorem (some shortest walk is not a path)
• derives contradiction
• states contradiction means assumption false
• all logic-based claims are true
• all graph-based claims are true
• appeals to definition of path (no repeat vertex)
• appeals to shortness/length in some way

quiz 12 – no key released; students were permitted to take up to 2 pages, replacing previous grades if they did so.

final quiz – no key released; students were permitted to take up to all pages, replacing previous grades if they did so.