Problems 01-25
1 (a) A proposition, True ($\text{T}$). (b) A proposition, $\text{T}$. (c) A proposition, $\text{T}$. (d) A proposition, False $\text{F}$. (e) Not a proposition, $\text{T}$ when $x = 2$, $\text{F}$ otherwise. (f) Not a proposition, because not a declarative sentence.
2 (a) Not a proposition, because not a declarative sentence. (b) Not a proposition, because is a question, i.e. not a declarative sentence. (c) A proposition, very very likely $\text{T}$ (d) Not a proposition, but if $x = 1$, then $\text{T}$. (e) A proposition, $\text{F}$. (f) Not a proposition, because the sentence is mathematical statement with variables.
3 (a) If
$$ \phantom{,} \ p = \text{``Linda is younger than Sanjay''} \ , $$then
$$ \phantom{,} \ \lnot p = \text{``Linda is not younger than Sanjay''} \ . $$(b) If
$$ \phantom{,} \ p = \text{``Mei makes more money than Isabella''} \ , $$then
$$ \phantom{,} \ \lnot p = \text{``Mei makes less money than Isabella''} \ . $$(c) If
$$ \phantom{,} \ p = \text{``Moshe is taller than Monica''} \ , $$then
$$ \begin{array}{r c l} \lnot p & = & \text{``Moshe is not taller than Monicaj} \\ & = & \text{``Moshe is shorter than Monica''} \ . \end{array} $$(d) If
$$ \phantom{,} \ p = \text{``Abby is richer than Ricardo''} \ , $$then
$$ \phantom{,} \ \lnot p = \text{``Abby is poorer than Ricardo''} \ . $$4
(a) $\lnot p = \text{``Janice has less Facebook friends than Juan.''}$
(b) $\lnot p = \text{``Quincy is not smarter than Venkat.''}$
(c) $\lnot p = \text{``Zelda drives less miles to school than Paola.''}$
(d) $\lnot p = \text{``Briana wakes up earlier than Gloria.''}$
5
(a) $\lnot p = \text{``Mei doesn't have a MP3 player.''}$
(b) $\lnot p = \text{``There's pollution in New Jersey.''}$
(c) $\lnot p = \text{``}2 + 1 \neq 3\text{.''}$
(d) $\lnot p = \text{``The summer in Maine is not hot and sunny.''}$
6
(a) $\lnot p = \text{``Jennifer and Teja aren't friends.''}$
(b) $\lnot p = \text{``There aren't $13$ items in a baker's dozen''}$
(c) $\lnot p = \text{``Abby sen less than $100$ text messages yesterday.''}$
(d) $\lnot p = \text{``$121$ is not a perfect square.''}$
7
(a) $\lnot p = \text{``Steve does not have more than $100$ GB free disk space on his laptop.''}$
(b) $\lnot p = \text{``Zach does not block e-mails or texts from Jennifer.''}$
(c) $\lnot p = \text{``$7 \cdot 11 \cdot 13 \neq 999$.''}$
(d) $\lnot p = \text{``Diane didn't rode her bicycle 100 miles on Sunday.''}$
8
(a) Let $\operatorname{N}_{\text{RAM}}(\text{X})$ denote the number of random access memory in a cell phone. Then the proposition can be expressed as
$$ \begin{array}{r c l} (\operatorname{N}_{\text{RAM}}(\text{B}) > \operatorname{N}_{\text{RAM}}(\text{A})) & \land & (\operatorname{N}_{\text{RAM}}(\text{B}) > \operatorname{N}_{\text{RAM}}(\text{C})) \\[0.5em] & \Updownarrow & \\[0.5em] (288 > 128) & \land & (288 > 256) \\[0.5em] & \Updownarrow & \\[0.5em] \text{T} & \land & \text{T} \\[0.5em] & \Updownarrow & \\[0.5em] & \phantom{.} \text{T} , \end{array} $$which means that the proposition is true.
(b) The sentence can be broken down into proposition
and
The conjunction of these proposition is therefore
$$ \phantom{,} \ p \lor q \ . \tag{1} $$Since $\text{Smartphone C}$ has more ROM than $\text{Smartphone B}$, i.e. $p = \text{T}$, and $\text{Smartphone C}$ has lower resolution than $\text{Smartphone B}$, i.e. $q = \text{F}$, the proposition $(1)$ is $\text{T} \lor \text{F} = \text{T}$.
(c) Similarily as above, the propositions are:
Then the main proposition can be written as
$$ \phantom{.} \ p \land q \land r \ . $$Evalutating the truth values means that $p$ is false ($\text{F}$), $q$ is true ($\text{T}$), and $r$ is false ($\text{F}$), so the main proposition
$$ \phantom{.} \ \phantom{.} \ p \land q \land r = \text{F} \land \text{T} \land \text{F} \ . $$In order for the logical conjunction to be true, all truth values should be true, so the proposition is $\text{F}$.
(d) The propositions are:
so the proposition ginve by the task can be written as
$$ \phantom{.} \ (p \land q) \rightarrow r \ . $$Writin out the truthtable for the propositions
$$ \begin{array}{ | c | c | c | c | c | } \hline \quad p \quad & \quad q \quad & \quad r \quad & \quad p \land q \quad & \quad (p \land q) \rightarrow r \quad \\[0.5em] \hline \text{T} & \text{T} & \text{T} & \text{T} & \text{T} \\[0.5em] \text{T} & \text{T} & \text{F} & \text{T} & \text{F} \\[0.5em] \text{T} & \text{F} & \text{T} & \text{F} & \text{T} \\[0.5em] \text{T} & \text{F} & \text{F} & \text{F} & \text{T} \\[0.5em] \text{F} & \text{T} & \text{T} & \text{F} & \text{T} \\[0.5em] \text{F} & \text{T} & \text{F} & \text{F} & \text{T} \\[0.5em] \text{F} & \text{F} & \text{T} & \text{F} & \text{T} \\[0.5em] \text{F} & \text{F} & \text{F} & \text{F} & \text{T} \\[0.5em] \hline \end{array} $$Since $p$ is $\text{T}$, $q$ is $\text{T}$, and $r$ is $\text{F}$, it can be seen from the second row of the above truth table that the proposition of the task is $\text{F}$.
(e) The propositions of the declaration are
so the statement can be written as
$$ p \leftrightarrow q $$which is equal to
$$ \phantom{.} \ (p \rightarrow q) \land (q \rightarrow r) \ . $$The truth table for this expression is
$$ \begin{array}{ | c | c | c | c | c | } \hline \quad p \quad & \quad q \quad & \quad p \rightarrow q \quad & \quad q \rightarrow p \quad & \quad (p \rightarrow q) \land (q \rightarrow p) \quad \\[0.5em] \hline \text{T} & \text{T} & \text{T} & \text{T} & \text{T} \\[0.5em] \text{T} & \text{F} & \text{F} & \text{T} & \text{F} \\[0.5em] \text{F} & \text{T} & \text{T} & \text{F} & \text{F} \\[0.5em] \text{F} & \text{F} & \text{T} & \text{T} & \text{T} \\[0.5em] \hline \end{array} $$Because $p$ is $\text{F}$ and obviously $q$ is $T$, then it can be seen from the 3rd row of the above truth table that the statment if $\text{F}$.
9 (a) The proposition is $\text{F}$. (b) True and true, so the proposition is $\text{T}$. (c) False and false, so $\text{F}$. (d) The first proposition is $\text{F}$, the second $\text{T}$, so the proposition is $\text{T}$ by the definition of implication. (e) Because $p \leftrightarrow q = (p \rightarrow q) \land (q \rightarrow p)$, $p = \text{F}$ and $q = \text{T}$, the proposition is $\text{T}$.
10 When the propositions of the declaration are
$\qquad q =$ “I won the million dollar jackpot”.
thegiven (compound) propositions are:
(a) $\lnot p =$ “I didn’t buy a lottery ticket this week”.
(b) $p \lor q =$ “I gought a lottery ticket this week of I won the million dollar jackpot”
(c) $p \rightarrow q =$ “If I bought a lottery ticket this week, then I won the million dollar jackpot”.
(d) $p \land q =$ “I bought a lottery ticket this week and I won the million dollar jackpot.”
(e) $p \leftrightarrow q =$ “If and only if I bought a lottery ticket this week, I won the million dollar jackpot”.
(f) $\lnot p \rightarrow \lnot q =$ “If I didn’t buy a lottery ticket this week, then I did not win the million dollar jackpot”.
(g) $\lnot p \land \lnot q =$ “I did not buy a lottery ticket this week and I did not win the million dollar jackpot”. ticket this week, then I did not win the million dollar”.
(h) $\lnot p \land (p \land q) =$ “I did not buy a lottery ticket this week or I bought a lottery ticket this week and I won the million dollar jackpot”.
11 When the propositions of the declaration are
$\qquad q =$ “Sharks have been spotted near the shore”.
the given (compound) propositions are:
(a) $\lnot q =$ “Swimming at the New Jersey shore is not allowed”
(b) $p \land q =$ “Swimming at the New Jersey shore is allowed and sharks have been spotted near the shore”.
(c) $\lnot p \land q =$ “Swimming at the New Jersey shore is not allowed or sharks have been spotted near the shore”
(d) $p \rightarrow \lnot q =$ “If swimming at the New Jersey shore is allowed, then sharks have not been spotted near the shore”.
(e) $\lnot q \rightarrow p =$ “If sharks have not been spotted near the shore, then sharks have been spotted near the shore”
(f) $\lnot p \rightarrow \lnot q =$ “If swimming at the New Jersey shore is not allower, then sharks have not been spotted near the shore”.
(g) $p \leftrightarrow \lnot q =$ “If and only if swimming at the New Jersey shore is allowed, sharks have not been spotted near the shore”.
(h) $\lnot p \land (p \lor \lnot q) =$ “If swimming at the New Jersey shore is not allowed and swimming at the New Jersey shore is allowed or sharks have not been spotted near the shore”.
12 When the propositions of the declaration are
$\qquad q =$ “The votes have been counted”.
the given (compound) propositions are:
(a) $\lnot p =$ “The election is not decided”.
(b) $p \land q =$ “The election is decided and the votes have been counted”.
(c) $\lnot p \land q =$ “The election is not decided and the votes have been counted”.
(d) $q \rightarrow p =$ “If the election is decided, then the votes have been counted”.
(e) $\lnot q \rightarrow \lnot p =$ “If the votes have not been counted, then the election is not decided”.
(f) $\lnot p \rightarrow \lnot q =$ “ If the election is not decided, then the votes have not been counted”.
(e) $p \leftrightarrow q =$ “If and only if the election is decided, the votes have not been counted”.
(f) $\lnot q \land (\lnot p \lor q) =$ “The votes have not been counted or the election is not decided and the votes have been counted”.
13 When the propositions of the declaration are
$\qquad q =$ “It is snowing”.
the propositions are:
(a) $p \land q$. (b) $p \land \lnot q$. (c) $\lnot p \land \lnot q$. (d) $q \leftrightarrow p$ (e) $p \rightarrow q$. (f) $(p \lor q) \lor (p \rightarrow q)$ (g) $p \leftrightarrow q$.
14 When the propositions of the declaration are
$\qquad q =$ “You miss the final examination”.
$\qquad r =$ “You pass the course”.
then
(a) $p \rightarrow q =$ “If you have the flu, you miss the final examination”.
(b) $\lnot q \rightarrow r =$ “If you don’t have the flu, then you pass the course”.
(c) $q \rightarrow \lnot r =$ “If you have the flu, then you don’t pass the course”.
(d) $p \land q \land r =$ “You have the flu, (or) you miss the final examination, or you pass the course”.
(e) $(p \rightarrow \lnot r) \lor (q \rightarrow \lnot r) =$ “If you have the flu, then you don’t pass the course, or if you miss the final examination, then you don’t pass the course”.
(f) $(p \land q) \lor (\lnot q \land r) =$ “You have the flu and you miss the final examination or you don’t miss the final examination anr you pass the course”.
15 When the propositions of the declaration are
$\qquad q =$ “You get a speeding ticket”.
then
(a) $\lnot q$. (b) $p \rightarrow \lnot q$. (c) $q \rightarrow p$. (d) $\lnot p \rightarrow \lnot q$. (e) $p \rightarrow q$. (f) $q \rightarrow \lnot p$. (g) $q \leftrightarrow p$.
16 When the propositions of the declaration are
$\qquad q =$ “You do every exercise in this bookticket”.
$\qquad r =$ “You get an A in this class”.
then
(a) $p \rightarrow \lnot q$. (b) $p \land q \land r$. (c) $r \rightarrow p$. (d) – (e) $(p \land q) \rightarrow r$. (f) $r \rightarrow (q \lor p)$.
17 When the propositions of the declaration are
$\qquad q =$ “Hiking is safe on the trail”.
$\qquad r =$ “Berries are ripe along the trail”.
then
(a) $r \land \lnot p$. (b) $p \land q \land r$. (c) $r \rightarrow (q \leftrightarrow p)$. (d) $\lnot q \land \lnot p \land r$. (e) $(\lnot q \land \lnor) \rightarrow q$. (f) $(p \land r) \rightarrow \lnot q$.
18 The truth table for biconditional, given in the book, is
$$ \begin{array}{ | c | c | c | } \hline p & q & p \leftrightarrow q \\ \hline \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{F} \\ \text{F} & \text{F} & \text{T} \\ \hline \end{array} $$(a) then “$2 + 2 = 4$ if and only if $1 + 1 = 2$” is true ($\text{T}$) because both mathematical expressions are true.
(b) then “1 + 1 = 2 if and only if $2 + 3 = 4$” is false ($\text{F}$) because $2 + 3 = 5$, not $4$, so the right hand-side of the biconditional operator is $\text{F}$ making the whole proposition $\text{F}$.
(c) then “$1 + 1 = 3$ if and only if monkeys can fly” is $\text{F}$ because $1 + 1 \neq 3$, but $2$, so the left hand-side of the biconditional is $\text{F}$ making the whole proposition $\text{F}$.
(d) then “$0 > 1$ if and only if $2 > 1$”, is $\text{T}$ because both sides of the biconditional are $\text{F}$.
19 Truth table for the logical implication is
$$ \begin{array}{ | c | c | c | } \hline p & q & p \rightarrow q \\ \hline \text{T} & \text{T} & \text{T} \\ \text{T} & \text{F} & \text{F} \\ \text{F} & \text{T} & \text{T} \\ \text{F} & \text{F} & \text{T} \\ \hline \end{array} $$so
(a) the statement “If $1 + 1 = 2$”, then $2 + 2 = 5$ is $\text{F}$, because the $2 + 2 \neq 5$, but $4$, and $1 + 1 = 2$.
(b) from the statement “If $1 + 1 = 3$”, then $2 + 2 = 4$, it can be seen that $1 + 1 \neq 3$, but $2$, so it’s $\text{F}$. The right hand-side is $\text{T}$. Because of these, the whole statement is $\text{T}$.
(c) the statement “if $1 + 1 = 3$”, then $2 + 2 = 5$, is $\text{T}$, because both propositions are $\text{F}$.
(d) the statement “if monkeys can fly, then $1 + 1 = 3$” is $\text{T}$, for the same reason as is (c), that is, both propositions in the statement are false, assuming that no flying monkeys exist.
20 Based on the truth table of implication given in 19,
(a) “If $1 + 1 = 3$, then unicorns exist” is true ($\text{T}$), because $1 + 1 \neq 3$, since $1 + 1 = 2$.
(b) “If $1 + 1 = 2$, then $2 + 3 = 4$” is false ($\text{F}$), because it is true that $1 + 1 = 2$, but it’s not true that $2 + 3 = 4$, because $2 + 2 = 4$.
(c) “If $ 1 + 1 = 2$, then dogs can fly” is $\text{F}$, because even if the first part of the implication is true, the second part very likely isn’t because no flying dogs shouldn’t exits.
(d) “If monkeys can fly, then $1 + 1 = 3$”, is $\text{F}$ because $1 + 1 \neq 3$, $1 + 1 = 2$, and no flying monkeys should exist.
21
(a) The proposition most likely is the situation of exclusive or ($\text{XOR}$), even though in the proposition “Coffee of tea comes with dinner” the part “but not both” is not implied. In these kind of situations, it is common that $\text{XOR}$ is assumed.
(b) The proposition is a situation of inclusive or ($\text{OR}$), because digits are characters, so one can choose a situation of both, that is, one can have a password of eight characters that has three digits.
(c) The proposition is a situation of $\text{OR}$, because usually one could have done a course in number theory or a course in cryptography, but it is still a possibility that one could have done both courses too.
(d) The proposition is a situation of $\text{XOR}$, or should be, because it’s common that a payment for a good or a service is required only once, so one shouldn’t need to pay, say first in euros and then in dollars or vice versa.
22
(a) The proposition seems to imply that either experience with C++ or Java is needed, but not necessarily both, so the proposition has a $\text{OR}$.
(b) Based on common experience, usually in these kind of situation, one can pick one or the other, but not both, so the proposition is based on $\text{XOR}$.
(c) One can have both of the document, i.e. either passport or voter registration card, so $\text{OR}$.
(d) Most likely a $\text{XOR}$, because it shouldn’t be possible to do both, “publish or perish”.
23
(a) If the sentence means inclusive or $\text{OR}$ or exclusive or ($\tex{XOR}$), then in the first meaning the sentence means that one needs needs to have calculus background or computer science course taken, but not both. In the case of exclusive or ($\text{XORJ}$), one needs both to take the discrete mathematics course.
(b) In the case of $\text{OR}$, it is possible to get both goods, i.e. the returned money and the loan. In the case of $\text{XOR}$, one can have on or the other, but not both, when buying a new car.
(c) In the case of $\text{OR}$, one can have either two, three or a total five items. In the case of $\text{XOR}$, one can have either two or three.
(d) Sentence means that if there are either two feet of snow, the wind chill falls below $\sim 100 ^\circ \text{F}$, or both, the school is closed ($\text{OR}$). On the other hand, it could be that if there are two feet of snow or the wind chill falls below $\sim 100 ^\circ \text{F}$, the school is open, but if both happen at the same time, the school is closed $\text{XOR}$.
24
(a) If one washed boss’s car, then one gets promoted.
(b) If wind blows from the south, then a spring thaw.
(c) If you bought the computer less than a year ago, then the warranty is good.
(d) If Willy cheats, then Willy gets caught.
(e) If you access the webpage, then you pay a subscription fee.
(f) If one knows the right people, then one gets elected.
(g) If Carol is on a boat, Caro gets sick.
25
(a) If the wind blows from the northeast, then it snows.
(b) If it stays warm for a week, then the apple trees will blossom.
(c) If the Pistons win the championship, then they beat the Lakers.
(d) If one wants to get to the top of Long’s Peak, then one needs to walk eight miles.
(e) If one wants to get tenure as a professor, then one needs to be world famous.
(f) If you dirve more than $400$ miles, then you will need to buy gasoline.
(g) If your guarantee is good, then you bought your CD player less than 90 days ago.
(h) If you begin your climb when it’s not too late, then you will reach the summit.
(i) If people believe in science, then we will have a future.