Question

Decide whether or not the following are statements. In the case of a statement, say if it is true or false, if possible. Either \(x\) is a multiple of 7 , or it is not.

Short answer

The sentence 'Either \(x\) is a multiple of 7, or it is not' is a true mathematical statement.

Step-by-step solution

Identify the Statement Type

The sentence 'Either \(x\) is a multiple of 7, or it is not' is presented in a format indicating it could be a mathematical statement. It suggests that \(x\) can only exhibit one of two qualities: being a multiple of 7 or not being a multiple of 7.

Determine If the Statement Is True or False

Given that any number \(x\) indeed must either be a multiple of 7 or not, the statement holds true for every possible value of \(x\). Therefore, the statement is always true.

Confirm the Nature of the Statement

Based on earlier identification and test for truth, one can confirm that this sentence is a statement, and it's invariably true.

Key concepts

Logical Reasoning

Logical reasoning is the underpinning of mathematics and is used to deduce conclusions based on given premises. It is the process of using a rational, systematic series of steps to conclude or to derive a solution from known facts. Taking our exercise as an example, the sentence 'Either x is a multiple of 7, or it is not' is subjected to logical reasoning to determine its nature as a mathematical statement.

In this case, logical reasoning helps in identifying that the given sentence provides a condition that cannot be false under any circumstances—it's a tautology. Each value of x is tested against the condition of being a multiple of 7 or not. This either-or scenario leverages the law of excluded middle in logic which states that a proposition must either be true or false, and there is no middle ground, thereby making the reasoning process binary and straightforward.

Truth Values in Mathematics

Truth values are fundamental in mathematics; they are the basis of all logical statements that can be classified as true or false. In our exercise, the determination of the statement's truth value is crucial. The statement 'Either x is a multiple of 7, or it is not' is always true, as there exists no number for which the statement doesn't hold.

This binary setup is reflective of the concept of bivalence in classical logic, which implies that any proposition must be either true or false. Thus, revealing the truth value of a statement in mathematics isn't just a formality but a foundational step in evaluating its validity and subsequently engaging in various forms of logical operations such as proofs, hypothesis testing, or decision making.

Logic in Mathematics

Logic in mathematics is a precise tool for reasoning, which involves systematic methods of arriving at a conclusion. It's the backbone of developing arguments, solving problems, and proving theorems. When we say a sentence such as 'Either x is a multiple of 7, or it is not' is a statement, we're employing mathematical logic. This statement is an example of a biconditional where both conditions cannot be true simultaneously, and neither can they be false at the same time.

Understanding mathematical logic means being able to differentiate between statements, propositions, and various types of logical connectives. It also means being familiar with concepts such as quantifiers and the structure of logical proofs. The use of logic in this scenario simplifies complex problems and guides mathematicians to draw conclusions that are universally accepted within the scope of the mathematical world.