Discrete Math ALL
In Java, the proposition ¬𝑝 is denoted by -p ~p !p \p #tags# in java, the proposition not p is denoted by
!p
8.1 Determine the edge whose removal makes the graph bipartite [image 81-007] Enter the edge within parenthesis with no spaces: e.g. (3,5)
(2,6) [or] (6,2)
8.1 Determine the edge whose removal makes the graph bipartite [image 81-003] Enter the edge within parenthesis with no spaces: e.g. (3,5)
(4,2) [or] (2,4)
Let the universal set be 𝑈={1,2,3,...,10}, and let 𝐴={1,4,7,10}, 𝐵={1,2,3,4,5}, and 𝐶={2,4,6,8}. List the elements of (𝐴∪𝐵)−(𝐶−𝐵). Write all the elements in increasing order, separated by commas (e.g. 2,3,4,7)
1,2,3,4,5,7,10
Let the universal set be U={1,2,3,4,5,6,7,8,9,10} and let A={1,4,7,10} B={1,2,3,4,5} and C={2,4,6,8}. List the elements of (A∪B') ∩ (B∪C). Write all the elements in increasing order, separated by commas (e.g. 2,3,4,7)
1,4,6,8
5.2 Convert 85 sub 10 to binary form. 85:2=42 remainder 1 42:2=21 remainder 0 21:2=10 remainder 1 10 :2=5 remainder 0 5 :2=2 remainder 1 2 :2=1 remainder 0 1 :2=0 remainder 1 Thus 88sub10 = 1011000 sub2
1010101
5.4 Assume that we choose primes p = 29, q = 53, and n = 233. Apply the RSA cryptosystem guidelines to compute: z = 𝜙 = s =
1537 1456 25
Let the universal set be U={1,2,3,...,10}, and let A={1,4,7,10}, B={1,2,3,4,5}, and C={2,4,6,8}. List the elements of C−(A∪B'). Write all the elements in increasing order, separated by commas (e.g. 2,3,4,7)
2
8.1 A 5-cube has been labeled with 5-digit binary labels. Determine the length of the shortest path connecting vertex 00110 with vertex 01011
3
The cardinality of {𝑎,{𝑎,𝑏},𝑏} is. 2 4 none of the above 3 #tags#
3
The cardinality of {𝑎,𝑏,{𝑎,𝑏},𝑏} is. 3 2 4 none of the above
3
Let the universal set be U={1,2,3,...,10}, and let A={1,4,7,10} B={1,2,3,4,5} and C={2,4,6,8}. List the elements of (A∩B)−(B−C). Write all the elements in increasing order, separated by commas (e.g. 2,3,4,7)
4
5.4 Quiz 5.4 A01 Encrypt 485 using an RSA cryptosystem with public key (1711, 297).
483
Let the universal set be U={1,2,3,...,10}, and let A={1,4,7,10}, B={1,2,3,4,5}, and C={2,4,6,8}. List the elements of (A−C)−(A∩B). Write all the elements in increasing order, separated by commas (e.g. 2,3,4,7)
7,10
Let the universal set be U={1,2,3,4,5,6,7,8,9,10}, and let A={1,4,7,10}, B={1,2,3,4,5}, and C={2,4,6,8}. List the elements of (A−C)−(A∩B). Write all the elements in increasing order, separated by commas (e.g. 2,3,4,7)
7,10
Which of the following sentences is not a proposition
Ask Mark if he is done with his homework
Which of the following sentences is not a proposition Tomorrow I will take a test Rational numbers don't exist This is my cat Ask Mark if he is done with his homework
Ask Mark if he is done with his homework
A coin is flipper 10 times. What is the negation of the proposition: "More heads than tails were obtained"? At least five tails were obtained Less heads than tails were obtained No heads or no tails were obtained More tails than heads were obtained
At least five tails were obtained
A coin is flipper 10 times. What is the negation of the proposition: "More heads than tails were obtained"? More tails than heads were obtained Less heads than tails were obtained At least five tails were obtained No heads or no tails were obtained
At least five tails were obtained
1. Test 1 - Part 1 The proposition 𝑝→𝑞 is logically equivalent to ¬𝑝→¬𝑞 True False #tags# p implies q is logically equivalent
False
Assuming that 𝑝 and 𝑟 are false and that 𝑞 is true, find the truth value of ¬𝑝→¬𝑞∨𝑟 True False #tags#
False
Assuming that 𝑝 is true, while 𝑞 and 𝑟 are false, find the truth value of (𝑞→𝑝)∧(𝑝→𝑟) True False #tags#
False
If 𝑝 is false, 𝑞 is true, while 𝑟's status is unknown at this time, what is the truth value of 𝑝∧(¬𝑟→𝑞)? False Unknown True #tags#
False
If 𝑝 is false, 𝑞 is true, while 𝑟's status is unknown at this time, what is the truth value of 𝑝∧(¬𝑟→𝑞)? Unknown True False #tags#
False
If 𝑝 is the proposition "This game was fun" and 𝑞 is the proposition "I won this game", then ¬(𝑝∨𝑞) is the proposition "This game was fun but I did not win it" True False
False
If 𝑝 is the proposition "This game was fun" and 𝑞 is the proposition "I won this game", then ¬(𝑝∨𝑞) is the proposition "This game was not fun or I did not win it" True False #tags# not (p or q)
False
Test 2 Part 2 Assuming that 𝑝 and 𝑟 are false and that 𝑞 and 𝑠 are true, find the truth value of (𝑞∨(¬𝑠→¬𝑝))→(𝑝∨𝑟) #tags# (q or (not s implies not p))
False
Test 2 Part 2 A02 Assuming that 𝑝 and 𝑟 are false and that 𝑞 and 𝑠 are true, find the truth value of ((𝑞∧𝑠)→¬𝑟)∧(𝑝→(¬s→𝑞)
False
The cardinality of {∅} is 0. True False #tags# a set containing empty set is 0 a set containing null set is 0
False
The cardinality of ∅ is 1. True False #tags# null set is 1 empty set is 1
False
The following sets are equal: A={x∈R∣x2+x=2} B={−1,1} True False
False
The following sets are equal: A={x∈R∣x2−x−2=0} B={−1,0,2} True False
False
The following sets are equal: 𝐴={𝑥∈𝑅∣𝑥2+𝑥=2}𝐵={−1,1} True False
False
The following sets are equal: 𝐴={𝑥∈𝑅∣𝑥2−3𝑥+2=0} 𝐵={1,3} True False
False
The following sets are equal: 𝐴={𝑥∈𝑅∣𝑥2−𝑥−2=0} 𝐵={−1,0,2} True False
False
The following sets are equal: 𝐴={𝑥∈𝑅∣𝑥2−𝑥−2=0}𝐵={−1,0,2} True False
False
5.4 Encrypt the message UNDER_THE_BOOK using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: ORWFKUZHEPLTVXDQ_YJGBIMSNCA (use only capital letters and use _ for space)
IDKUJOBEUOWQQT
1. Test 1 - Part 1 Which of the following sentences is not a proposition #tags# michael are you home
Michael, are you home?
5.4 Encrypt the message ARRIVING_TODAY using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: ORWFKUZHEPLTVXDQ_YJGBIMSNCA (use only capital letters and use _ for space)
RJJPMPDHOBQKRC
A group of students is talking about their summer vacation. What is the negation of the proposition: "All students did not go on vacation"? Not all student went on vacation No students did go on vacation Some students went on vacation All students went on vacation
Some students went on vacation
Test 3 Part 2 A1 Q1 Write the truth table of (𝑝∨𝑞)∧(¬𝑝∨𝑞)∧(𝑝∨¬𝑞) #tags#
T F F F
Write the truth table of ¬(𝑝∧¬𝑞)∨(¬𝑝∧𝑞) #tags#
T F T T
Write the truth table of (𝑝∨𝑞)∧(¬𝑝∨𝑞)∨(𝑝∧¬𝑞)
T T T F
Write the truth table of (𝑝∨𝑞)∨(¬𝑝∧𝑞)
TFFT TFFT TTTT FTFF
Write the truth table of (𝑝∧𝑞)∨(¬𝑝∨𝑞)
TFTT FFFF FTTT FTTT
Write the truth table of ¬(𝑝∨𝑞)∨(𝑝∧𝑞) #tags#
TFTT TFFF TFFF FTFT
Encrypt the message ARRIVING_TODAY using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: LTEBJZSQYD_VMWOIRUXCHKPFANG (use only capital letters and use _ for space)
TXXDPDOQLHIJTN
If 𝑝 and 𝑞 are proposition, by 𝑝∧𝑞 we denote The disjunction of p and q The conjunction of p and q The intersection of p and q The union of p and q #tags# by p and q we denote
The conjunction of p and q
If 𝑝 and 𝑞 are proposition, by 𝑝∧𝑞 we denote The union of p and q The conjunction of p and q The disjunction of p and q The intersection of p and q #tags# p and q we denote
The conjunction of p and q
If 𝑝 and 𝑞 are proposition, by 𝑝∨𝑞 we denote The conjunction of p and q The intersection of p and q The union of p and q The disjunction of p and q #tags# by p or q we denote
The disjunction of p and q
If 𝑝 is a proposition, by ¬𝑝 we denote The disjunction of p The opposite of p The negation of p The complement of p #tags# if p is a proposition, by not p we denote
The negation of p
? Test 3 Part 2 A1 Q2 Assuming that 𝑝 and 𝑟 are false and that 𝑞 and 𝑠 are true, find the truth value of (𝑝↔𝑞)→(𝑟↔𝑠)
True
Assuming that 𝑝 and 𝑟 are false and that 𝑞 and 𝑠 are true, find the truth value of ((𝑞∧𝑠)→¬𝑟)∧((𝑝→(𝑟∨𝑞))∧𝑠) True False #tags#
True
Assuming that 𝑝 and 𝑟 are false and that 𝑞 and 𝑠 are true, find the truth value of ((𝑞∧𝑠)→¬𝑟)∧(𝑝→(¬𝑠→𝑞)) True False #tags#
True
Assuming that 𝑝 and 𝑟 are false and that 𝑞 and 𝑠 are true, find the truth value of (𝑝↔𝑞)→(𝑟↔𝑠) True False #tags#
True
Assuming that 𝑝 and 𝑟 are false and that 𝑞 is true, find the truth value of (¬𝑝→¬𝑞)∨¬𝑟 True False #tags# not p implies not q
True
Assuming that 𝑝 and 𝑟 are false and that 𝑞 is true, find the truth value of (𝑝∧𝑟)→𝑞 True False #tags#
True
If 𝑝 is false, 𝑞 is true, while 𝑟's status is unknown at this time, what is the truth value of 𝑝∨(¬𝑟→𝑞)? Unknown False True #tags#
True
If 𝑝 is the proposition "This game was fun" and 𝑞 is the proposition "I won this game", then ¬(𝑝∨𝑞) is the proposition "It is not true that this game was fun or I won it" True False
True
If 𝑝 is the proposition "This game was fun" and 𝑞 is the proposition "I won this game", then 𝑝∨𝑞 is the proposition "This game was fun or I won it" True False #tags# p or q is the proposition this game was fun or i won it p or q is the proposition "this game was fun or i won it"
True
The cardinality of {∅} is 1. True False #tags# null set is 1 empty set is 1
True
The cardinality of ∅ is 0. True False #tags# null set is 0 empty set is 0
True
The following sets are equal: A={1,2,2,3}B={1,2,3} True False
True
The following sets are equal: A={1,2,2}B={2,2,1} True False
True
The following sets are equal: 𝐴={1,1,3} 𝐵={3,3,1} True False
True
The following sets are equal: 𝐴={1,2,2,3}𝐵={1,2,3} True False
True
The following sets are equal: 𝐴={1,2,2} 𝐵={2,2,1} True False
True
The following sets are equal: 𝐴={{1,1,2},2} 𝐵={{1,2},2,2}
True
The following sets are equal: 𝐴={{1,1,2},2}𝐵={{1,2},2,2} True False
True
The following sets are equal: 𝐴={𝑥∈𝑅∣𝑥2+𝑥=2} 𝐵={−2,1} True False
True
The following sets are equal: 𝐴={𝑥∈𝑅∣𝑥2−𝑥−2=0}𝐵={−1,2} True False
True
If 𝑝 is false, 𝑞 is true, while 𝑟's status is unknown at this time, what is the truth value of 𝑝∨(𝑟→¬𝑞)? Unknown False True #tags#
Unknown
If 𝑝 is false, 𝑞 is true, while 𝑟's status is unknown at this time, what is the truth value of 𝑝∨𝑟→¬𝑞? False True Unknown #tags#
Unknown
5.4 Decrypt the message XUEEUHQKFHQN__F using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: QVLIAHWSFUBJZCE_NDMOKYTXPRG (use only capital letters and use _ for space)
WINNIE_THE_POOH
5.4 Encrypt the message TURN_TO_THE_RIGHT using the key defined as character: _ABCDEFGHIJKLMNOPQRSTUVWXYZ replace by: LBMOREZSPKACQD_IHNUVYTFJGXW (use only capital letters and use _ for space)
YTU_LYILYPELUKSPY
In 𝑝→𝑞, 𝑝 is referred as the (check all that applies) conclusion consequent antecedent hypothesis #tags# p implies q, p is referred as the p implies q, p is referred to as the
antecedent hypothesis
A group of students is talking about their summer vacation. What is the negation of the proposition: "More than two students went on vacation"? more than two students did not go on vacation less than two students did not go on vacation less than two students went on vacationvacation less than three students went on vacation
less than three students went on vacation
A coin is flipped 10 times. What is the negation of the proposition: "Some heads and some tails were obtained"? no heads or no tails were obtained some heads or some tails were obtained some tails and some heads were obtained all heads and all tails were obtained
no heads or no tails were obtained
In Java, the proposition ion 𝑝∧𝑞 is denoted by p | q p && q p || q p & q #tags# p and q is denoted by
p && q
In Java, the proposition ion 𝑝∨𝑞 is denoted by p V q p !! q p || q p ! q #tags# p or q is denoted by
p || q
In Java, the proposition 𝑝∨𝑞 is denoted by p !! q p ! q p V q p || q #tags# in java, the proposition p or q is denoted by
p || q
The symbol Q represents the set of complex numbers real numbers integer numbers rational numbers
rational numbers
Test 4 Part 1 Study Guide A02 Given an equivalence relation R on a set X, for any 𝑥∈𝑋, by [𝑥] we denote
the equivalence class of x in R
5.4 Assume that we choose primes p = 31 , q = 37, and n = 77. Apply the RSA cryptosystem guidelines to compute: z = 𝜙 = s =
z = 1147 𝜙 = 1080 s = 533
5.4 Assume that we choose primes p = 29, q = 41, and n = 47. Apply the RSA cryptosystem guidelines to compute: z = 𝜙 = s =
z = 1189 𝜙 = 1120 s = 143
5.4 Assume that we choose primes p = 23, q = 59, and n = 355. Apply the RSA cryptosystem guidelines to compute: z = 𝜙 = s =
z = 1357 𝜙 = 1276 s = 895
Let 𝑋 and 𝑌 be subsets of a universal set 𝑈. The set 𝑋−𝑌 is defined as {𝑢∈𝑈∣𝑢∉𝑋 and 𝑢∉𝑌} {𝑢∈𝑌∣𝑢∉𝑋} none of the above {𝑢∈𝑈∣𝑢∈𝑋 and 𝑢∉𝑌}
{𝑢∈𝑈∣𝑢∈𝑋 and 𝑢∉𝑌}
The cardinality of a set 𝑋 is denoted by
|𝑋|
1. Test 1 - Part 1 The second De Morgan's Law is ¬(𝑝∧𝑞)≡¬𝑝∧¬𝑞 ¬(𝑝∧𝑞)≡(¬𝑝∧¬𝑞) ¬(𝑝∨𝑞)≡(¬𝑝∨¬𝑞) ¬(𝑝∧𝑞)≡¬𝑝∨¬𝑞 #tags# de morgans law
¬(𝑝∧𝑞)≡¬𝑝∨¬𝑞
The first De Morgan's Law is ¬(𝑝∨𝑞)≡¬𝑝∨¬𝑞 ¬(𝑝∨𝑞)≡(¬𝑝∨¬𝑞) ¬(𝑝∧𝑞)≡¬𝑝∧¬𝑞 ¬(𝑝∨𝑞)≡¬𝑝∧¬𝑞 #tags# de morgans law
¬(𝑝∨𝑞)≡¬𝑝∧¬𝑞
The contrapositive of 𝑝→𝑞 is ¬𝑞→¬𝑝 𝑝→¬𝑞 𝑞→𝑝 ¬𝑝→𝑞 #tags# the contrapositive of p implies q is
¬𝑞→¬𝑝
4.1-4.3a Test 4 Part 1 Study Guide A02 We say that 𝑓(𝑛) is Ω(𝑔(𝑛)) when there exists a constant 𝐶1 such that |𝑓(𝑛)| [Select] 𝐶1|𝑔(𝑛)| so all but finitely many positive integers.
≥
Which of the following is incorrect? 𝑄𝑛𝑜𝑛𝑛𝑒𝑔⊆𝑅 𝑄+⊆𝑅 𝑄⊆𝑅𝑛𝑜𝑛𝑛𝑒𝑔 𝑄+⊆𝑅𝑛𝑜𝑛𝑛𝑒𝑔
𝑄⊆𝑅𝑛𝑜𝑛𝑛𝑒𝑔
Test 3 Part 1 A1 Q1 The proposition 𝑝 if and only if 𝑞 is denoted by 𝑝⇔𝑞 𝑝↔𝑞 𝑝≡𝑞 𝑝=𝑞
𝑝↔𝑞
Let 𝑝: "There is a hurricane" 𝑞: "It is raining" Which of the following represents "There is a hurricane but it is not raining" 𝑝∨¬𝑞 𝑝∧¬𝑞 p and not q ¬(𝑝∨𝑞) ¬(𝑞∧𝑝)
𝑝∧¬𝑞
Let 𝑝: "There is some coffee" 𝑞: "We have breakfast" Which of the following represents "There is some coffee but we don't eat breakfast" ¬(𝑞∧𝑝) 𝑝∧¬𝑞 ¬(𝑝∨𝑞) 𝑝∨¬𝑞
𝑝∧¬𝑞
Let 𝑝: "There is some coffee" 𝑞: "We have breakfast" Which of the following represents "There is some coffee but we don't eat breakfast" 𝑝∧¬𝑞 p and not q 𝑝∨¬𝑞 p or not q ¬(𝑝∨𝑞) not (p or q) ¬(𝑞∧𝑝) not (q and p)
𝑝∧¬𝑞
Let 𝑝: "There is some coffee" 𝑞: "We have breakfast" Which of the following represents "Either there is some coffee or we don't eat breakfast" 𝑝∧¬𝑞 ¬(𝑝∨𝑞) ¬(𝑞∧𝑝) 𝑝∨¬𝑞
𝑝∨¬𝑞
In 𝑝→𝑞, the conclusion is → 𝑞 𝑝 none of the above #tags# in p implies q, the conclusion is in p implies q the conclusion is
𝑞
In 𝑝→𝑞, the consequent is none of the above → 𝑞 𝑝 #tags# in p implies q the consequent is
𝑞
