94 Quantifiers [40] Suppose that the students at Memorial University all live in either St. Johns or Mount Pearl. Let J be the set of students in St. Johns, and P be the set of students in Mount Pearl. Then the set of all students $S = J \cup P$. We define \texttt{class} as a Boolean function taking two people as arguments, which returns true if they are in a class together and false if they are not. Also, the order of arguments does not matter: $\forall x, y \in S \cdot \text{class}(x, y) \iff \text{class}(y, x)$ (a) Express the following natural language sentences formally, using quantifiers, sets, etc. [10 marks per part, 20 total] (i) Every student from Mount Pearl is in a class with somebody from St. Johns. (ii) There is at least one student from St. Johns who is not in any classes with a student from Mount Pearl. (b) Express the following formal sentences in clear English. [10 marks per part, 20 total] (i) $\forall x \in J \cdot \exists y \in J \cdot (x = y) \land \text{class}(x, y)$ (ii) $\exists x \in J \cdot \forall y \in P \cdot \text{class}(x, y)$
Added by Kim M.
Close
Step 1
Step 1: Read through the text carefully to check for any spelling or typographical errors. Show more…
Show all steps
Your feedback will help us improve your experience
Sri K and 68 other AP CS educators are ready to help you.
Ask a new question
Labs
Want to see this concept in action?
Explore this concept interactively to see how it behaves as you change inputs.
Key Concepts
Recommended Videos
Determine the truth value of each expression below if the domain is the set of all real numbers. ∃x∀y (xy = 0) (If true, give an example.) ∀x∀y∃z (z = (x - y)/3) (If false, give a counterexample.) ∀x∀y (xy = yx) (If false, give a counterexample.) ∃x∃y∃z (x^2 + y^2 = z^2) (If true, give an example.) Redo the above (problem 1), with the domain of positive integers. Translate each of the following English statements into logical expressions. The domain of discourse is the set of all integers. There are two numbers whose sum is equal to their product. The product of every two positive integers is positive. Every positive integer can be expressed as the sum of the squares of four integers. There is a positive integer that is smaller than all other positive integers. The domain of discourse is the members of a chess club. The predicate B(x, y) means that person x has beaten person y at some point in time. Give a logical expression equivalent to the following English statements. No one has ever beaten Nancy. Everyone has been beaten before. Everyone has won at least one game. No one has beaten both Ingrid and Dominic. There are two members who have never been beaten. Translate each of the following English statements into logical expressions. The domain of discourse is the set of all real numbers. The reciprocal of every positive number is positive. There is no smallest number. There are two numbers whose ratio is less than 1. Write the negation of each of the following logical expressions so that all negations immediately precede predicates. ∀x ∃y ∃z P(y, x, z) ∃x ∃y P(x, y) ∧ ∀x ∀y Q(x, y) ∃x ∀y ( P(x, y) ↔ P(y, x) ) ∃x ∀y ( P(x, y) → Q(x, y) ) Use a truth table to prove the conclusion from the hypotheses. The hypotheses are: If I drive on the freeway, I will see the fire. I will either drive on the freeway or take surface streets. I am not going to take surface streets. Conclude that I will see the fire. Use the following variable names: p: I drive on the freeway r: I take surface streets q: I see the fire p q r Use the laws of logic to prove the conclusion from the hypotheses. Give propositions and predicate variable names in your proof. Use the set of all students as the domain of discourse. The hypotheses are: Larry and Hubert are taking Boolean Logic. Any student who takes Boolean Logic can take Algorithms. Conclude that Larry and Hubert can take Algorithms. Use the laws of logic to prove the conclusion from the hypotheses. Give propositions and predicate variable names in your proof. Use the set of all people as the domain of discourse. The hypotheses are: Everyone who practices hard is a good musician. There is a member of the orchestra who practices hard. Conclude that someone in the orchestra is a good musician. Which of the following arguments are valid? Explain your reasoning. I have a student in my class who is getting an A. Therefore, John, a student in my class is getting an A. Every girl scout who sells at least 50 boxes of cookies will get a prize. Suzy, a girl scout, got a prize. Therefore Suzy sold 50 boxes of cookies. Use the laws of logic to show that ∀x(P(x) ∧ Q(x)) implies that ∀x Q(x) ∧ ∀x P(x).
Sri K.
Problem 9: (Second Proof That Two Sets Are Equal) Let I = (5, 9]. Consider the sets T = {x | x is a lower bound of I} and B = {x | x ≤ 5}. Prove that T = B. An important set that will show up often throughout the semester is the set with nothing in it, which we call the empty set. Definition (Empty Set) The empty set is the set ∅ = {} that contains no elements. If we think of a set as a box with elements in it, then the empty set is a box with nothing in it. Here is another axiom that you have probably used many times in your life without ever realizing it. Axiom (Well Ordering Principle) Every nonempty subset S of the natural numbers has a least element. By least element, we mean that there is a natural number m which is an element of S such that m ≤ x for every x in S. Problem 10: (Which Dominoes Remain Standing) Suppose that Jon has set up an infinite number of dominoes, with the dominoes numbered 1, 2, 3, .... The dominoes are set up so that if the kth domino falls, then the (k + 1)st domino will also fall. So if the 7th domino falls, then the 8th must fall as well. Jon knocks down the first domino, which starts causing other dominos to fall. Which dominos fall? Which dominoes remain standing? Make sure you prove your result. The well ordering principle will come in handy. Suggestion: Use set builder notation to help you, so let F = {n ∈ N | domino n fell} and S = {n ∈ N | domino n remains standing}. Then make some claims and prove they are correct. Definition (Statement And Open Sentence) • A statement is a sentence that can be classified as either true or false (but not both). The truth value of a statement is either "True" or "False." For a sentence to be a statement, it is not necessary that we know the truth value, but it must be clear that the value is either "True" or "False." • Some sentences involve a variable, and the truth value of the sentence cannot be determined until the value of the variable is specified. An open sentence is a sentence involving a variable whose truth value cannot be determined until the variables in the sentence are specified, at which point the open sentence becomes a statement.
Formalize the following argument by using the given predicates and then rewriting the argument as a numbered sequence of statements. Identify each statement as either a premise, or a conclusion that follows according to a rule of inference from previous statements. In that case, state the rule of inference and refer by number to the previous statements that the rule of inference used. Lions hunt antelopes. Ramses is a lion. Ramses does not hunt Sylvester. Therefore, Sylvester is not an antelope. Predicates: H(x,y)=" x hunts y", L(x)="x is a lion" and A(x)="x is an antelope". The domain of discourse is all animals. Prove that there can be no perfect square between 25 and 36, i.e. there is no integer n so that 25 < n^2 < 36. Prove this by directly proving the negation. Your proof must only use integers, inequalities and elementary logic. You may use that inequalities are preserved by adding a number on both sides, or by multiplying both sides by a positive number. You cannot use the square root function. Do not write a proof by contradiction.
Recommended Textbooks
Computer Science and Information Technology
Introduction to Programming Using Python
Computer Science - An Overview
Transcript
18,000,000+
Students on Numerade
Trusted by students at 8,000+ universities
Watch the video solution with this free unlock.
EMAIL
PASSWORD