= Opgaver
#image("Opgaverne.png")
== Opgave 1.1-1
#image("Opgave-1.1-1.png")
a) Proposition. And it is true

b) Proposition. And it is false.

c) Proposition. True

d) Proposition. False

e) Not a proposition because it has a variable.

f) Not a proposition because it is a command.

== Opgave 1.1-3
#image("Opgave 1.1-3.png")
a) Linda is not younger (older or same age) than Sanjay

b) Mei does not make more money than Isabella (makes less or same)

c) Moshe is not taller than Monica (shorter or same age)

d) Abby is not richer than Ricardo


== Opgave 1.1-11
#image("Opgave-1.1-11.png")
a) Sharks have not been spotted near the shore.

b) Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore.

c) Either swimming is not allowed at the New Jersey shore or sharks have been spotted near the shore.

d) Swimming is allowed at the shore which implies sharks have not been spotted.

e) Sharks have not been spotted at the shore which implies it is allowed to swim there.

f) Swimming is not allowed at the shore which implies sharks have not been spotted.

g) Swimming is allowed at the shore if and only if sharks have not been spotted.

h) Swimming is allowed or sharks have not been spotted and swimming is not allowed.

== Opgave 1.1-37

#image("Opgave 1.1-37.png")
#table(columns: 10)[p][q][$not p$][$not q$][$p->not q$][$not p <->q$][$(p->q)or(not p->q)$][$(p->q)and(not p-> q)$][$(p<->q)or(not p<->q)$][$(not p <-> not q)<->(p<->q)$][T][T][F][F][F][F][T][T][T][T][F][T][T][F][T][T][T][T][T][T][T][F][F][T][T][T][T][F][T][T][F][F][T][T][T][F][T][F][T][T]


== Opgave 1.1-53
#image("Opgave 1.1-53.png")
a) Only the 99th statement would be true. For example the 3rd statement cant be true because if only 3 of the statements are false, that would mean the other 97 statements are true, but they cant be because they say other amounts of statements are false. But for the 99th, it would be true that all the other statements are false except for that one. The 100th statement cant be true because if all statements in the list are false, then the 100th would be true, which it cant be if all are false.

b) Only the statements of $n<=50$ would could be true, because for the 51th statement "At least 51 of the statements are false" it is incorrect that _at least_ 51 of the statements are false (statements 1-50 are also true, so there's only 49 remaining false statements which is contradicting the statement).

c) It would mean that statements 1-49 would be true as for the 49th statement, at least 49 of the statements _are true_ (real answer is 50 but 50 is at least 49).

== Opgave 1.4-1
#image("Opgave 1.4-1.png")
a) $P(0)=T$

b) $P(4)=T$

c) $P(6)=F$

== Opgave 1.4-31
#image("Opgave 1.4-31.png")
a) $forall y Q(0,0,0) and forall y Q(0,1,0)$

b) $exists x Q(0,1,1) or exists x Q(1,1,1) or exists x Q(2,1,1)$

c) $exists z not Q(0,0,0) or exists z not Q(0,0,1)$

d) $exists x not Q(0,0,1) or exists x not Q(1,0,1) or exists x not Q(2,0,1)$

== Opgave 1.4-35
#image("Opgave 1.4-35.png")
a) $exists x(x<=1)$

b) $exists x(x>2)$

c) $forall x(x<4)$

d) $forall x(x>=0)$

e) $exists x((x>=-1) and (x<=2))$

f) $forall x((x>=4) and (x<=7))$

== Opgave 1.4-37

#image("Opgave 1.4-37.png")
a) No counterexample, it would only be happen for numbers between 0 and 1 (0.1,0.2,...0.9)

b) $exists x(x=0)$

c) $exists x(x>1or x<1)$

== Opgave 1.4-41
#image("Opgave 1.4-41.png")
a) If there exists a printer that is out of service and is also busy, that leads to some work being lost.

b) If all printers are busy, then a job is in the queue

c) If there exists a print job that gets queued but also lost, then a printer is out of service

d) If all printers are busy and all print jobs are queued, a print job has been lost


== Opgave 1.4-55
#image("Opgave 1.4-55.png")
a) Truth

b) False

c) True