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321 Section, Week 2

321 Section, Week 2. Natalie Linnell. Extra Credit problem. All lions are fierce Some lions do not drink coffee Some fierce creatures do not drink coffee. Translate into logic (provide defs for predicates). Ax(P(x)->Q(x)), Ex(P(x)/-R(x)), Ex(Q(x)/-R(x). Discussion. Discuss:

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321 Section, Week 2

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  1. 321 Section, Week 2 Natalie Linnell

  2. Extra Credit problem

  3. All lions are fierceSome lions do not drink coffeeSome fierce creatures do not drink coffee • Translate into logic (provide defs for predicates) Ax(P(x)->Q(x)), Ex(P(x)/\-R(x)), Ex(Q(x)/\-R(x)

  4. Discussion Discuss: Why A->? Why not A/\? Why E/\? Why not E->?

  5. All lions are fierceSome lions do not drink coffeeSome fierce creatures do not drink coffee • Negate all the statements Ex(P(x)/\-Q(x)), Ax(-P(x)vR(x)), Ax(-Q(x)vR(x))

  6. All lions are fierceSome lions do not drink coffeeSome fierce creatures do not drink coffee • Argue whether the reasoning to conclude the third statement from the first two is sound Yes; 2existential instationiation;1 universal instantiation

  7. All hummingbirds are richly coloredNo large birds live on honeyBirds that do not live on honey are dull in colorHummingbirds are small • Translate into logic (define any predicates used) Ax(P(x)->S(x)), -Ex(Q(x)/\R(x)), Ax(-R(x)->-S(x)), Ax(P(x)->-Q(x))

  8. All hummingbirds are richly coloredNo large birds live on honeyBirds that do not live on honey are dull in colorHummingbirds are small • Negate all the statements Ex(P(x)/\-S(x)), Ex(Q(x)/\R(x)),Ex(-R(x)/\S(x)), Ex(P(x)/\Q(x))

  9. All hummingbirds are richly coloredNo large birds live on honeyBirds that do not live on honey are dull in colorHummingbirds are small • Argue whether the fourth statement follows from the first 3 Yes; 1, 3 contrapositive and modus ponens, 2(modus ponens if in ->; else -/\ means if one true other false)

  10. Show that xP(x)  xQ(x) is not equivalent to x(P(x)  Q(x)) • Let P(x) and Q(x) be statements from math or the world to illustrate this. P(x)=x is positive, Q(x)=x is negative; any two contradictory statements would do

  11. There is a student in this class who has been in every room of at least one building on campus • Translate into logic (define any predicates) ExEyAz(P(z,y)->Q(x,y))

  12. Every student in this class has been in at least one room of every building on campus • Translate into logic (define any predicates) AxAyEz(P(z,y)/\Q(x, z))

  13. xy (xy = y) • Domain is real numbers – what concept does this capture? Multiplicative identity

  14. xy (((x < 0)  (y<0))  (xy>0)) • Domain is real numbers – what concept does this capture? Product of two negative integers is positive.

  15. Everyone has exactly one best friend • Translate into logic – do not use uniqueness quantifier AxEy(B(x,y)/\Az(x!=y)->-B(x,z)

  16. It is not sunny this afternoon and it is colder than yesterdayWe will go swimming only if it’s sunnyIf we do not go swimming, then we will take a canoe tripIf we take a canoe trip, then we will be home by sunset • Show that “We will be home by sunset” follows 1.5 eg 6

  17. Negate xy P(x, y) v xy Q(x, y)

  18. Negate xy (P(x, y)  Q(x,y))

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