Prposition logic | Computer Science homework help

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Question 1

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Which of these statements are true, for propositional logic? (In an exam you would have to justify your answers).

Select one or more:

A. If a formula is not satisfiable then it is not valid

B. X is not satisfiable if and only if ¬X is valid

C. If a formula is not valid then it is not satisfiable

D. X is not valid if and only if not X is satisfiable

Question 2

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For each propositional formula below, construct a truth table. Which formulas are valid?

Select one or more:

A. (prightarrow p)

B. p

C. (pwedge q)

D. ((prightarrow(qvee r)) leftrightarrow((prightarrow q)vee(prightarrow r)))

E. (prightarrow(qrightarrow p))

Question 3

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In each case say if the formula is satisfiable.

Select one or more:

A. neg(prightarrow(qrightarrow p))

B. p

C. neg((prightarrow q)rightarrow p)

D. (pwedge q)

E. (pwedgeneg p)

Question 4

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Which of the following sets of connectives are functionally complete for propositional logic?

Select one or more:

A. wedge, vee, neg

B. rightarrow, bot (false)

C. vee, neg

D. wedge, vee

E. rightarrow, wedge

Question 5

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Which of the following are propositional formulas, according to the strict definition of propositional formulas?

Select one or more:

A. neg(p)

B. prightarrow q

C. (pwedge q)

D. ((prightarrow q)rightarrow p)

E. (pwedge qwedge r)

Question 6

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Consider the following 11 propositional formulas

  1. (prightarrow(qrightarrow p))

  2. (q rightarrow p)

  3. ( neg p vee  q )

  4. (neg p wedge neg q )

  5. (p vee neg p )

  6. (p vee neg q )

  7. ((p veeneg q) wedge (neg p vee q))

  8. (p wedgeneg p)

  9. (p rightarrow  q)

  10. ((p wedge neg q) vee (neg p wedge  q))

  11. (p leftrightarrow  q)

Which of these eleven formulas are equivalent to each other. Choose one from the following:

Select one:

A. 1=5, 2=3, 7=11, 4=10, 6=9

B. None of the other answers are right

C. 1=5, 2=6, 3=9, 7=11

D. 1=5, 2=6, 4=7=10, 3=9

E. None are equivalent

Question 7

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Which of the following propositional formulas are in disjunctive normal form?

Select one or more:

A. neg p

B. (p veeneg q)

C. ((p veeneg q) wedge r)

D. ((p wedge q) vee (neg p wedgeneg q))

E. ((neg p wedge q) vee (p wedge neg q))

Question 8

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Which of the following statements is true?

Select one or more:

A. There is a DNF formula which is equivalent to all possible propositional formulas.

B. There is no DNF formula equivalent to (p wedgeneg p)

C. For every propositional formula there is a CNF formula equivalent to it.

D. For every propositional formula there is a DNF formula equivalent to it.

Question 9

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Let i be the propositional valuation where i(p) = t, i(q) = t, i(r) = f, …

Let v be the truth function that extends i. Which of the formulas below evaluate to true under this valuation v?

Select one or more:

A. (((p leftrightarrow q) rightarrowneg(p wedgeneg r)) veeneg r )

B. (neg p rightarrow (q wedgeneg p))

C. (p wedgeneg r)

D. (neg p rightarrow q)

Question 10

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Let L be a first order language with just one predicate, =, and no constants or function symbols. Let An be a sentence that is true in a structure M if and only if M has at least n points in its domain. What is the smallet number of variables required to write such a sentence An?

Select one:

A. 2

B. n

C. n-1

D. 1

E. infinity

Question 11

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Let S=({mathbb N}, I) where I(<^2) is the set of all (x, y) where x is strictly less than y, constants 0, 1 denote zero and one respectively.  Which of the following first order formulas are true in the structure S?

Select one or more:

A. forall xexists y <^2(x, y)

B. neg (<^2(0, 1)rightarrow(0=+^2(0, 1)))

C. <^2(1, +^2(1, 0))

D. forall xexists y <^2(y, x)

E. (<^2(1, +^2(0, 0))vee (1=+^2(0, 1)))

Question 12

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Let S be the structure ({mathbb N}, I)  where the domain is the set of natural numbers and  I(<) is the set of pairs (x, y) where x is strictly less than y. Using S and the assignments A1 to A5 below, say which of the following are true.

A1:

x -> 7

y -> 14

z -> 9

w -> 5 (all other vars w)

 

A2:

x -> 8

y -> 7

z -> 9

w -> 5 (all other w)

 

A3:

x -> 0

y -> 14

z -> 9

w -> 5 (all other w)

 

A4:

x -> 8

y -> 14

z -> 9

w -> 5 (all other w)

 

A5:

x -> 6

y -> 14

z -> 9

w -> 5 (all other w)

Select one or more:

A. S, A1 |= small exists x <^2(x, 1)

B. S, A1 |= small forall xexists y <^2(x, y)

C. S, A3 |= small <^2(x, 1)

D. S, A2 |= small <^2(x, 1)

E. S, A2 |= small negexists z(<^2(y, z)wedge <^2(z, x))

Question 13

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Let L be a first-order language with just = as a predicate and no constants or function symbols. How many variables to you need to express a sentence that is true in a model if and only if the domain has exactly n elements?

Select one:

A. 2

B. n+1

C. n

D. n-1

E. 2n+1

Question 14

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In the following formula > means greater than, = means equals, * means times. Which statement below is a good translation of the first order formula?

small forall x[neg(exists yexists z(x=y*zwedge y>1wedge z>1))rightarrowexists w(w>xwedgeneg(exists yexists z(w=y*zwedge y>1wedge z>1)))]

 

Select one:

A. for every composite number there is a prime number

B. for every prime number there is a bigger prime number

C. x and w are prime numbers

D. all numbers bigger than x are prime.

E. for all x, if x is a prime number then w is a prime number.

Question 15

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Consider the first order formula:

small (forall x(exists y P^2(x, y)rightarrow R^2(y, x))rightarrow Q^1(x))

Which statements are correct?

Select one or more:

A. The scope of small forall x is small ((exists y P^2(x, y)rightarrowexists x R^2(y, x))rightarrow Q^1(x))

B. the scope of small exists y is small P^2(x, y)

C. small R^2(y, x) is in the scope of small exists x and small exists y, but not in small forall x.

D. there is one free occurence of small x: the small x in small Q^1(x)

E. This is not a well-formed formula.

Question 16

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Take a first order language with constants C = {0,1}, predicates P = {R^2} and functions F = {+^2, -^1, times^2}.

Which of the following are terms in this language?

Select one or more:

A. +^2(x, y, 1)

B. times^2(+^2(0, 1), +^2(0, 1))

C. R^2(x, 0)

D. -^1(0, 1)

E. +^2(3, 0)

Question 17

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Let S be the structure ({mathbb N}, I)  where I(<) is the set of pairs (x, y) where x is strictly less than y.Which of these first order formulas are valid in S?

Select one or more:

A. exists xforall y(<^2(x, y)vee (x=y))

B. forall yexists x(<^2(x, y)vee (x=y))

C. forall xforall y((<^2(x, y)vee <^2(y, x))vee x=y)

D. forall yexists x <^2(y, x)

E. exists xforall y <^2(y, x)

Question 18

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Which of these first order formulas are valid?

Select one or more:

A. (forall xneg R^1(x) rightarrownegexists x R^1(x))

B. (exists xforall y <^2(x, y)rightarrow forall yexists x <^2(x, y))

C. forall xforall y ((<^2(x, y)vee <^2(y, x))vee(x=y))

D. (forall xexists y <^2(x, y) rightarrow exists yforall x <^2(x, y))

Question 19

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Predicate Logic. Consider the following assignments.

A1:

x -> 7

y -> 14

z -> 9

w -> 5 (all other vars w)

 

A2:

x -> 8

y -> 7

z -> 9

w -> 5 (all other w)

 

A3:

x -> 0

y -> 14

z -> 9

w -> 5 (all other w)

 

A4:

x -> 8

y -> 14

z -> 9

w -> 5 (all other w)

 

A5:

x -> 6

y -> 14

z -> 9

w -> 5 (all other w)

 

Which statements are correct?

Select one or more:

A. A1 is an x-variant of A3

B. A5 is a z-variant of A5

C. A4 is a z-variant of A5

D. A2 is a y-variant of A4

E. A3 is an x-variant of A5

Question 20

 

Question text

Let S be the structure ({mathbb N}, I)  where I(<) is the set of pairs (x, y) where x is strictly less than y, I(+) is the ordinary addition function, I(0), I(1) are the integers zero, one respectively..Using the structure S  calculate the interpretation of

+2(+2(1,1), +2(0,1))

Answer:

Question 21

Let S be the structure ({mathbb N}, I)  where I(<) is the set of pairs (x, y) where x is strictly less than y. Let A be the assignment where x -> 5 and y -> 8.

Calculate [+ 2(x, y)]S,A

Answer:

 

 

 

 

 

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