Translate each of these statements into logical expressions...

Translate each of these statements into logical expressions in three different ways by varying the domain and by using predicates with one and with two variables. a) Someone in your school has visited Uzbekistan. b) Everyone in your class has studied calculus and $\mathrm{C}++.$ c) No one in your school owns both a bicycle and a motorcycle. d) There is a person in your school who is not happy. e) Everyone in your school was born in the twentieth century.

Translate each of these statements into logical expressions in three different ways by varying the domain and by using predicates with one and with two variables.
a) Someone in your school has visited Uzbekistan.
b) Everyone in your class has studied calculus and $\mathrm{C}++.$
c) No one in your school owns both a bicycle and a motorcycle.
d) There is a person in your school who is not happy.
e) Everyone in your school was born in the twentieth century.

Key Concepts

Logical Expressions

Logical expressions are symbolic representations used to denote statements in formal logic. These expressions use symbols, predicates, quantifiers, and connectives to encapsulate the meaning of natural language statements in a structured and precise way, allowing for rigorous analysis and reasoning.

Domain of Discourse

The domain of discourse refers to the set of entities over which variables in logical expressions range. Selecting an appropriate domain is crucial in translating natural language statements into logical forms, as it specifies the context and the objects that are being considered, such as all students in a school or all individuals in a class.

Predicate Logic

Predicate logic extends propositional logic by incorporating predicates and quantifiers, which enable statements about properties of objects and relationships among them. It allows for more complex translations of natural language, where predicates can involve one or more variables to encapsulate properties or relations relevant to the subject matter.

Quantifiers

Quantifiers, such as the existential (?) and universal (?) quantifiers, are used to express the scope of applicability of a predicate across elements in the domain. They are instrumental in logical translations of statements that involve notions of 'someone', 'everyone', 'no one', and other forms of generality or existence.

Logical Connectives

Logical connectives, including conjunction (and), disjunction (or), and negation (not), are used to build complex logical expressions by combining simpler predicates. They mirror the structure of natural language expressions, ensuring that compound statements are accurately represented in logical form.

Translation Strategies

Translation strategies involve varying both the domain of discourse and the arity of predicates (using one or two variables) to effectively capture the intended meaning of natural language statements. This flexibility allows different logical formulations that might be more suitable based on the context, such as focusing directly on individuals or on relations between them.

Transcript

00:01 In this problem, we are going to be focusing on quantifiers, where we're given a statement, and we have to find essentially the domain of that statement using our knowledge of mathematical quantifiers.

00:14 Now, quantifiers are very important, and they're going to come up and be significant when you get into introductory proofs and proofs later on.

00:24 But for right now, we're just getting used to the concept of quantifiers and how to use them and the difference between them.

00:31 So please, while you're going through this video, please look back at the problem to remind yourselves of the original statements that we were given.

00:40 I think that's going to help if you're new to quantifiers and learning it for the first time.

00:46 Okay.

00:47 So for part a, we're going to let the propositional function be y of x.

00:52 And we're going to say x is the school.

00:55 Then we're going to say we have this additional function u of x is that x has visited.

01:02 Uzbekistan.

01:04 So the domain of this, using these conditions, is that there exists y of x and u of x.

01:13 So what does that mean? there exists someone in the school who has visited uzbekistan.

01:19 Now suppose that a person x has visited a country y, which will denote b of x comma y.

01:27 So now when we take all of this into account, we can say that the domain of the original statement is that there exists y of x and a v of x comma uzbekistan.

01:43 Now let's move on to part b.

01:46 We're going to follow essentially the same procedure.

01:49 Let's consider functions c of x and p of x.

01:52 So the domain of just these functions at the moment would be for all x, we have x, pardon me, y of x, maps to c of x and p of x.

02:06 So suppose that there is a person x who studied subject y and will denote this by s comma, pardon me, s of x comma y.

02:16 So the domain of this, remember we're considering the grade and calculus, is that for all, all x, this would hold.

02:25 Y of x maps to s of x comma calculus and s comma x c plus plus.

02:38 Now on to part c, we're going to follow the same procedure.

02:42 So we're going to let b of x and m of x be functions...