Let S(x) be the predicate “x is a student,” F(x) the...

Let S(x) be the predicate “x is a student,” F(x) the predicate “x is a faculty member,” and A(x, y) the predicate “x has asked y a question,” where the domain consists of all people associated with your school. Use quantifiers to express each of these statements. a) Lois has asked Professor Michaels a question. b) Every student has asked Professor Gross a question. c) Every faculty member has either asked Professor Miller a question or been asked a question by Professor Miller. d) Some student has not asked any faculty member a question. e) There is a faculty member who has never been asked a question by a student. f ) Some student has asked every faculty member a question. g) There is a faculty member who has asked every other faculty member a question. h) Some student has never been asked a question by a faculty member.

Let S(x) be the predicate “x is a student,” F(x) the predicate “x is a faculty member,” and A(x, y) the predicate “x has asked y a question,” where the domain consists of all people associated with your school. Use quantifiers to express each of these statements.
a) Lois has asked Professor Michaels a question.
b) Every student has asked Professor Gross a question.
c) Every faculty member has either asked Professor Miller a question or been asked a question by Professor Miller.
d) Some student has not asked any faculty member a question.
e) There is a faculty member who has never been asked a question by a student.
f ) Some student has asked every faculty member a question.
g) There is a faculty member who has asked every other faculty member a question.
h) Some student has never been asked a question by a faculty member.

Key Concepts

Translation of Natural Language to Formal Logic

This process involves identifying the logical structure of natural language statements and expressing them using predicates, quantifiers, and logical connectives. It requires careful analysis to ensure that the formal expressions accurately reflect the intended meaning of the sentences.

Logical Connectives

Logical connectives like 'and' (?), 'or' (?), 'not' (¬), and 'implies' (?) are used to combine or modify predicates and quantifiers. They structure complex logical formulas by linking simpler components into more meaningful expressions.

Predicate

A predicate is a function that takes one or more subjects from the domain of discourse and returns a truth value. It represents properties or relationships about objects, such as 'is a student' or 'has asked a question', which are then used to build formal logical statements.

Quantifiers

Quantifiers allow statements to express assertions about collections of objects. The universal quantifier (?) is used to indicate that a predicate holds for all objects in the domain, while the existential quantifier (?) asserts that there is at least one object in the domain for which the predicate holds.

Domain of Discourse

The domain of discourse is the set of all objects under consideration in the logical analysis. This concept limits the range over which the predicates and quantifiers operate, such as all people associated with an institution in the context of the problem.

Transcript

00:01 Hey, it's clarissa enumerate here.

00:02 So we're given s of x, which is x as a student, f of x, which is x is a faculty member, and a of x, comma, y, which is x has asked y a question.

00:13 And the domain are all the people associated with the school.

00:17 So for a, we see that the given statement is the definition of a of x, y, with x being lowest, and y is michaels.

00:29 So we have a of lois comma professor michaels.

00:51 For part b, we can rewrite the statement as for all people, if there are a student, then they ask professor gross a question.

01:06 So this is, we use our universal for x, s of x, then a, which means x has asked y a question so it's x comma professor gross for part c we can rewrite the statement as for all people if they are a faculty member then they ask professor millen a question or professor miller asked them a question so our universal of f of x which is x is a faculty member then a which is as a faculty member then a which is as asked a question, professor miller, or professor miller asked them a question.

02:29 For part d, we can rewrite it as there is a person x who is a student and for all people, y.

02:37 If y is a faculty member, then this person x did not ask y a question.

02:43 So it's existential for x, s of x, which is x is a student, and university, and university, so for y, f of y, then not a of xy.

03:11 For part e, it says there is a faculty member who has never been asked a question by a student, which is there is a person x who is a faculty member whom all the people y, if they are a student, then y did not ask x a question.

03:30 So this is existential...

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