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: Why A->? Why not A/\? Why E/\? Why not E->?

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

All lions are fierce Some lions do not drink coffee Some 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

All hummingbirds are richly colored No large birds live on honey Birds that do not live on honey are dull in color Hummingbirds 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))

All hummingbirds are richly colored No large birds live on honey Birds that do not live on honey are dull in color Hummingbirds 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))

Argue whether the fourth statement follows from the first 3 All hummingbirds are richly colored No large birds live on honey Birds that do not live on honey are dull in color Hummingbirds are small Yes; 1, 3 contrapositive and modus ponens, 2(modus ponens if in ->; else -/\ means if one true other false)

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

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))

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))

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

 x  y (((x 0)) Domain is real numbers – what concept does this capture? Product of two negative integers is positive.

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

It is not sunny this afternoon and it is colder than yesterday We will go swimming only if it’s sunny If we do not go swimming, then we will take a canoe trip If 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

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

Negate  x  y (P(x, y)  Q(x,y))