X


Translating Mathematical Statements into Statements Involving Nested Quantifiers




Q. Translate the statement “The sum of two positive integers is always positive” into a logical expression.





Q. Translate the statement “Every real number except zero has a multiplicative inverse.” (A multiplicative inverse of a real number x is a real number y such that xy = 1).



Q. Translate the statement ∀x(C(x) ∨ ∃y(C(y) ∧ F(x,y))) into English, where C(x) is “x has a computer,” F(x,y) is “x and y are friends,” and the domain for both x and y consists of all students in your school.



Translate the statement ∃x∀y∀z((F (x, y) ∧ F(x,z)∧ (y ≠ z)) → ¬F(y,z)) into English, where F(a,b)means a and b are friends and the domain for x, y, and z consists of all students in your school.





Express the statement “If a person is female and is a parent, then this person is someone’s mother” as a logical expression involving predicates, quantifiers with a domain consisting of all people, and logical connectives.



Q. Express the statement “Everyone has exactly one best friend” as a logical expression involving predicates, quantifiers with a domain consisting of all people, and logical connectives.



Use quantifiers to express the statement “There is a woman who has taken a flight on every airline in the world.”






Rules of Inference




Consider the following argument involving propositions (which, by definition, is a sequence of propositions):
“If you have a current password, then you can log onto the network.”
“You have a current password.”
Therefore, “You can log onto the network.”
Determine whether this is a valid argument.





Definitions :



Rule of Inference - Modus ponens



Rule of Inference - Hypothetical syllogism



Example :State which rule of inference is used in the argument: If it rains today, then we will not have a barbecue today. If we do not have a barbecue today, then we will have a barbecue tomorrow. Therefore, if it rains today, then we will have a barbecue tomorrow.




Rule of Inference - Modus tollens





Rule of Inference - Disjunctive syllogism



Rule of Inference - Addition



Example :State which rule of inference is the basis of the following argument: “It is below freezing now. Therefore, it is either below freezing or raining now.”





Rule of Inference - Simplification



Example :State which rule of inference is the basis of the following argument: “It is below freezing and raining now. Therefore, it is below freezing now.



Rule of Inference - Conjunction





Rule of Inference - Resolution



Example : Use resolution to show that the hypotheses “Jasmine is skiing or it is not snowing” and “It is snowing or Bart is playing hockey” imply that “Jasmine is skiing or Bart is playing hockey.”



Example : Show that the premises (p ∧ q) ∨ r and r → s imply the conclusion p ∨ s.





Fallacy of affirming the conclusion


  1. The proposition ((p → q) ∧ q) → p is not a tautology, because it is false when p is false and q is true.

  2. However, the argument with premises p → q and q and conclusion p as a valid argument form although it is not.

  3. This type of incorrect reasoning is called the fallacy of affirming the conclusion.



Example Is the following argument valid? If you do every problem in a book, then you will learn discrete mathematics. You learned discrete mathematics. Therefore, you did every problem in this book.





Fallacy of denying the hypothesis



Q. If the conditional statement p → q is true, and ¬p is true, is it correct to conclude that ¬q is true? In other words, is it correct to assume that you did not learn discrete mathematics if you did not do every problem in the book, assuming that if you do every problem in this book, then you will learn discrete mathematics?


Previous Next