Set Theory Logic: Finding True Statements in a Given Domain

In summary, the conversation discusses two given open sentences and a domain, and the task is to determine all values of x in the domain for which the implication P(x) → Q(x) is true. The conversation also mentions the truth values for the implication operator and suggests that proving P is false would result in a true value for the implication. Ultimately, it is concluded that there are no values of x in the given domain for which P(x) is false, making the statement true vacuously.
  • #1
knowLittle
312
3

Homework Statement


In each of the two following open sentences P(x) and Q(x) over a domain S are given.
Determine all ##x \in S## for which P(x) → Q(x) is a true statement.

## P(x): x \in [-1, 2]; Q(x): x^{2} \leq 2; S=[-1,1] ##

Homework Equations


According to truth values for →:
a b a-> b
0 0 1
0 1 1
1 0 0
1 1 1

The Attempt at a Solution


If I can prove that P is False, then I will always get a T value for ->
Can I just say ## x \in [-1,1] ##, this would literally mean that statement P is false.
Could this count?
 
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  • #2
Would this mean that the statement is true vacuously?
 
  • #3
Is there any ##x \in S## such that P(x) is false?
 

FAQ: Set Theory Logic: Finding True Statements in a Given Domain

What is Set Theory?

Set theory is a branch of mathematics that deals with the study of sets, which are collections of objects or elements. It is used to understand mathematical concepts such as numbers, functions, and infinity.

What is the difference between a set and a subset?

A set is a collection of objects or elements, while a subset is a set that contains elements that are also in another set. In other words, all elements of a subset must also be present in the larger set.

What is the meaning of the symbols used in set theory, such as union (∪), intersection (∩), and complement (Aᶜ)?

The union of two sets A and B (A ∪ B) is the set of all elements that are in A or B (or both). The intersection of two sets A and B (A ∩ B) is the set of all elements that are in both A and B. The complement of a set A (Aᶜ) is the set of all elements that are not in A.

How is set theory used in other fields of study?

Set theory is used in various fields such as computer science, linguistics, and philosophy. In computer science, it is used to model data structures and algorithms. In linguistics, it is used to study the structure of language. In philosophy, it is used to understand logical reasoning and the foundations of mathematics.

What is the significance of set theory in mathematics?

Set theory is considered one of the foundations of mathematics and is used to define and study other mathematical concepts. It provides a rigorous framework for understanding mathematical structures and relationships between different objects. It is also used to prove mathematical theorems and solve problems in various branches of mathematics.

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