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In logic, a statement is a declarative sentence that can be classified as either true or false. Logical statements are fundamental to understanding logical reasoning and applications in mathematics and computer science.
For instance, take the statement 鈥淭he sky is blue.鈥� This is a logical statement because it can be identified as true or false. However, an imperative sentence like 鈥淐lose the door鈥� is not a logical statement because it does not say anything that can be classified as true or false.
In the given exercise, the statement is denoted as \(P(x)\) and it reads as 鈥淺(x \text{ is less than or equal to 4}\).鈥� By understanding logical statements, we can determine its truth value by substituting different values for \(x\) and evaluating whether \(P(x)\) turns out to be true or false.

Inequalities are expressions that show the relationship between two values or algebraic expressions. The basic forms of inequalities are: less than (<), greater than (>), less than or equal to (鈮�), and greater than or equal to (鈮�).
To evaluate an inequality, like \(x \text{ 鈮� 4}\), you have to check if a particular value of \(x\) satisfies the condition. If the inequality holds true, the value is a solution; otherwise, it is not.
In the exercise, the statement \(P(x)\) translates to \(x \text{ 鈮� 4}\). Let's evaluate it step-by-step:

• \(P(0)\): Substitute \(x\) with 0, forming \(0 \text{ 鈮� 4}\). This is true because 0 is indeed less than or equal to 4.
• \(P(4)\): Similarly, substitute \(x\) with 4, forming \(4 \text{ 鈮� 4}\). This is true because 4 equals 4.
• \(P(6)\): Substitute \(x\) with 6, forming \(6 \text{ 鈮� 4}\). This is false because 6 is greater than 4.

By understanding these simple steps, you can solve and understand various inequality problems.

Predicate logic extends propositional logic by including quantifiers and predicates that express properties of objects. Predicates are functions that return a true or false value given an input.
In the exercise, the predicate \(P(x)\) is defined as 鈥淺(x \text{ 鈮� 4}\).鈥� This means \(P(x)\) will return true for values of \(x\) that are less than or equal to 4, and false otherwise.
Predicate logic is useful in building more complex logical expressions. For example, if you have a set of numbers and you want to know which of them satisfy \(P(x)\), you use predicate logic to evaluate each number individually.
Understanding predicates and how to evaluate them is crucial, especially if you plan to dig deeper into logic, computing, and advanced mathematics.