Question

Translate each of the following sentences into symbolic logic. If \(\sin (x)<0,\) then it is not the case that \(0 \leq x \leq \pi\)

Short answer

The translation of the sentence into symbolic logic is \( (\sin (x)<0) \Rightarrow \neg (0 \leq x \leq \pi) \)

Step-by-step solution

Symbolize the Sentence

The given statement is an 'if, then' statement. In symbolic logic, such statements are usually represented as \( P \Rightarrow Q \) where \( P \) is the premise and \( Q \) is the conclusion. So, we can write the statement as: 'If \( P \) is \( \sin (x)<0 \), then \( Q \) is it is not the case that \(0 \leq x \leq \pi\) .'

Convert Conditions to Logical Symbols

The condition 'it is not the case that' is represented by the negation symbol in logic \(\neg\). Given that, our statement would then translate to: 'If \( P \) is \( \sin (x)<0 \), then \( Q \) is \(\neg (0 \leq x \leq \pi)\).'

Symbolize the Entire Statement

Now, we can symbolize the entire statement by combining the representations for both conditions. Therefore, the symbolized statement will be: \( (\sin (x)<0) \Rightarrow \neg (0 \leq x \leq \pi) \).

Key concepts

Mathematical Logic

Mathematical logic, a branch of mathematics and logic, involves the use of symbols to formally express ideas and arguments. It functions as a framework to convey precise meaning, removing ambiguity typically found in natural languages. The above exercise illustrates how mathematical logic can transform a conditional statement from plain English into a symbolic representation.
Logical statements, often made up of premises leading to a conclusion, can be evaluated to determine their truth values. In the exercise, this process involves analyzing a statement related to the sine function and its implications on the value range of the variable 'x'.
Understanding mathematical logic is fundamental for students as it underpins various areas of mathematics and computer science, helping to solve problems systematically and proving theorems conclusively.

Conditional Statements

Conditional statements are used to depict 'if-then' scenarios within logical reasoning. A conditional statement typically consists of two parts: a hypothesis (if) and a conclusion (then). In symbolic logic, they are written as \( P \Rightarrow Q \), where \( P \) and \( Q \) represent hypothetical propositions.
In the provided exercise, the statement \( \sin (x)<0 \) serves as the hypothesis \( P \), and the part stating 'it is not the case that \(0 \leq x \leq \pi\)' is the conclusion \( Q \). The exercise demonstrates how translating common language to logic symbols forces a meticulous examination of the meanings of the statements involved.
Mastering conditional statements encourages logical thinking and enhanced problem-solving skills as students learn to forge connections between different pieces of information.

Logic Symbols

Logic symbols are the basic vocabulary of symbolic logic, functioning as a shorthand form to represent logical operations and relations. Familiar ones include \(\land\) for 'and', \(\lor\) for 'or', \(eg\) for 'not', and \(\Rightarrow\) for 'implies'.
In our example, the negation symbol \(eg\) effectively denotes the denial of a condition, turning the phrase 'it is not the case that' into a concise, symbolic form. The usage of logic symbols, like those in the step by step solution, might seem straightforward, but each one holds a significant weight in the structure of the logic expressed.
By becoming familiar with logic symbols, students gain the ability to parse complex logical expressions and deduce the validity of arguments. This skill is particularly useful in computer science, philosophy, and more advanced mathematical courses.