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Write the following logical arguments as predicate expressions, defining the predicates used and domains of
variables. For each argument, mention the inference rules used in each step. [6 marks]
a) “Asim, a student in this class, knows how to write programs in Java. Everyone who knows how to write
programs in Java can get a high-paying job. Therefore, someone in this class can get a high-paying job.”
b) “Somebody in this class enjoys whale watching. Every person who enjoys whale watching cares about
ocean pollution. Therefore, there is a person in this class who cares about ocean pollution.”
c) “Each of the 50 students in this class has a laptop. Everyone who has a laptop can use a word processing
program. Therefore, Asim, a student in this class, can use a word processing program.”
d) “Everyone in Karachi lives within 50 miles of the ocean. Someone in Karachi has never seen the ocean.
Therefore, someone who lives within 50 miles of the ocean has never seen the ocean.”
What is the domain and range of the function f that assigns to each pair of negative integers x and y, the value x + y.
Prove or disprove: The average of three real numbers is greater than or equal to at least one of the numbers.
Prove or disprove : for every real number x, |x − 8| + x ≥ 6.
The federal Green Party recently elected a new leader: Bernie May. A party official said 23,877 Green voters cast a ballot in this race. If the voters were partitioned into 12 regions based on population, give a proof by contradiction that at least one region had at least 1990 voters.
Consider the following predicates over the domain of all dogs:
B(x): x barks
W(x): x wags its tail
S(x): x scratches the furniture
T(x): x wants a treat.
H(x): x goes to heaven
Translate the following propositions from simple English to logic or vice versa:
a) For every dog, if it scratches the furniture, then it does not wag its tail
b) There exists a dog that does not wag its tail but barks.
c) ∀x W (x) ∨ ∃x(B(x) ∧ ¬S(x))
d) ∃x (W (x) ↔ B(x))
e) There exists a dog that does not like treats or scratches on the furniture
f) All dogs go to heaven, provided they wag their tail and bark.
Given the following three statements, what can you conclude about James’s ability to climb a mountain? Justify your answer.
a. James is from Nova Scotia
b. No one who loves lobster can climb a mountain.
c. All people from Nova Scotia love lobster.
Let P (x, y) be a predicate. What is the negation of ∃x ∀y(P (x, y) ⇐⇒ P (y, x))?
1. Transform the following English statements into logical expressions. Identify the predicates, values, and their domain. Negate the logical expression and transform it back to English.
a. The raptors are better than every other team.
b. Every student either likes sharks or dolphins.
c. All students except one passed the final exam.
d. There is exactly a number that is bigger than 0.
No person lives underwater
Consider the following English statements and express as a logical expression with the same meaning.
The domain is the set of all students at Hogwarts.
H(x): x is a member of the Hufflepuff.
R(x): x is a member of Ravenclaw.
T(x,y): x takes a class with y.
a. There exists a student at Hogwarts
b. Every student is taking a class with Harry Potter
c. No student was placed in Hufflepuff
d. If a student took a class with Ron Weasley, then they did not take one with Harry Potter.
e. At least one student in Hufflepuff took a class with a student in Ravenclaw
Two students in Hufflepuff took a class with the same Ravenclaw student
#340153 on Dec 2023