Question

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

Short answer

Yes, 'The integer \(x\) is a multiple of 7' is a statement. But, without a specific value for \(x\), we cannot assign a truth value to it.

Step-by-step solution

Identify if Sentences are Statements

Firstly, check whether the given sentence can be classified as a statement. A statement must be a sentence that can either be true or false, but not both at the same time. Looking at 'The integer \(x\) is a multiple of 7', it can be observed that it is a sentence that can either be true or false.

Determining the Truth Value

In the second step, determine if the statement is true or false if it is possible. However, in this case, the truth of the sentence 'The integer \(x\) is a multiple of 7' depends on the value of \(x\). If \(x\) is a multiple of 7 (such as 7, 14, 21, 28, and so on), the statement is true. If \(x\) is not a multiple of 7 (such as 1, 2, 3, 8, and so on), the statement is false. Since \(x\) is not defined in the question, we are unable to definitively say whether the statement is true or false.

Key concepts

Truth Value

In the realm of mathematical logic, the truth value of a statement is a fundamental concept that refers to the truth or falsity of the propositions we encounter. A truth value indicates whether a statement is true or false and it is one of the most critical aspects to understand when delving into logical reasoning and mathematics. For instance, when examining the statement 'The integer x is a multiple of 7,' its truth value can vary. If x equals 14, a multiple of 7, the statement is true. Conversely, if x is 9, which is not a multiple of 7, the statement is false.

However, in situations where the value of x is not specified, the truth value cannot be determined. This undeclared variable yields the statement's truth value indeterminate. To establish truth values, one must have a clear understanding of the definition of the statement, the context in which it is used, and the values of any variables involved. In mathematics, acknowledging the conditionality of a statement's truth value is critical for higher-level reasoning and problem-solving.

Mathematical Logic

The backbone of clear reasoning in mathematics is mathematical logic, a systematic approach to evaluating arguments and statements. It encompasses a variety of elements, such as symbols, logical connectives, and quantifiers that collectively form the language used to convey mathematical truths effectively. The pursuit of mathematical logic involves not only analyzing statements and their truth values but also understanding how these statements can be combined or manipulated to form more complex arguments.

When it comes to exercises like 'Decide whether or not the following is a statement. If it is a statement, say if it is true or false if possible,' one employs mathematical logic to first identify the nature of the sentence given. Mathematical logic guides us to recognize the criteria for a logical statement - it must be something that can be assigned a truth value. This systematic process of evaluation helps students develop a structured approach to tackling mathematical problems and constructs a foundation for more advanced concepts in the field.

Statements in Mathematics

Understanding what constitutes statements in mathematics is foundational to mathematical discussion and reasoning. A statement is defined as a declarative sentence that is either true or false, but not both. It is a basic component that mathematicians use to build arguments, prove theories, and communicate ideas unambiguously. As seen in the textbook exercise, identifying if a given sentence is a mathematical statement is the first crucial step before any logical assessment can be made about its truth value.

The sentence 'The integer x is a multiple of 7' serves as a perfect example. This sentence is a mathematical statement because it meets the criteria: it asserts a claim that can be verified once the value of x is known. If it were unclear whether a sentence could be true or false (e.g., a question or a command), it would not qualify as a mathematical statement. An accurate understanding of statements in mathematics is paramount for evaluating hypotheses, solving equations, and proof construction.