COMP 382
Unit 1 Questions
Swarat Chaudhuri & John Greiner
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What do you need to do after class?
Do first assignment
Take first quiz
Sign up for tutorial section
Read 10 chapters of textbook
10
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List definition has how many cases?
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B.
C.
D.
function append (x: List, y: List): List {Response
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match x
case Empty => ???
case Cons(m, z) => ???
}
How to complete base case?
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function append (x: List, y: List): List {
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match x
case Empty => y
case Cons(m, z) => ???
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How to complete inductive case?
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function reverse (x: List): List {
match x
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How to complete inductive case?
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3
To prove: 𝑛 − 𝑛 is divisible by 3, for 𝑛 ∈ ℕ.
How many cases in inductive proof?
A.
B.
C.
D.
1
2
3
4
E. 𝑛 − 1
F. 𝑛
G. 𝑛 + 1
3
To prove: 𝑛 − 𝑛 is divisible by 3, for 𝑛 ∈ ℕ.
What is the inductive case?
A. If 𝑛3 − 𝑛 | 3, then
(𝑛 + 1)3 − 𝑛 + 1 | 3.
B. If (𝑛 + 1)3 − 𝑛 + 1 | 3, then
𝑛3 − 𝑛 | 3.
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𝑛
To prove: 𝑛! < 𝑛 , for 𝑛 > 1.
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What is base case?
A. 𝑛 = 0
B. 𝑛 = 1
C. 𝑛 = 2
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< is a well-founded relation over ℕ.
A. True
B. False
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 is a well-founded relation over ℕ.
A. True
B. False
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< is a well-founded relation over ℤ.
A. True
B. False
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Which is a well-founded relation on ℕ×ℕ?
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𝑎, 𝑏 < 𝑐, 𝑑 iff …
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$$
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𝑎<𝑐
𝑎 < 𝑐 and 𝑏 < 𝑑
𝑎 < 𝑐 or 𝑏 < 𝑑
𝑎 < 𝑐 or 𝑎 = 𝑐 and 𝑏 < 𝑑
𝑎+𝑏 <𝑐+𝑑
$$
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B.
C.
D.
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For chips puzzle, do induction on what?
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Remove a red and anything → Put in none.
Remove two yellow → Put in 1 yellow, 5 blue.
Remove a blue and not red → Put in 10 red.
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Total # of chips
Lex. order (#red, #yellow, #blue)
Lex. order (#blue, #yellow, #red)
Lex. order (#yellow, #blue, #red)
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B.
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Induction guarantees what for inductivelydefined programs?
Termination
Correctness
Most efficient algorithm
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Te
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