Problem 6 - A Gentle Introduction to the Art of Mathematics

Question

Sometimes in constructing a proof we find it necessary to "weaken" an inequality. For example, we might have already deduced that \(x

Step-by-Step Solution

Answer

Disjunctive Addition

Step 1 - Understand the Problem

Here, we need to determine the rule of inference being used when deducing the weakened inequality from a stronger one. Specifically, we need to understand how to deduce that if we know that \(x < y\), we can also conclude \(x \leq y\).

Step 2 - Define the Given and the Required

Given: \(x < y\) \ewline\ Required: \(x \leq y\)

Step 3 - Express the Inequality

Recall that \(x \leq y\) is equivalent to \(x < y \vee x = y\). This means \(x \leq y\) is true if either \(x < y\) is true or \(x = y\) is true.

Step 4 - Apply Disjunctive Addition

Using the rule of inference known as 'Disjunctive Addition,' we can infer \(x < y \vee x = y\) from the premise \(x < y\). Disjunctive Addition allows us to add any additional clause to a true statement to form a disjunction that remains true.

Step 5 - Conclusion

Therefore, by using Disjunctive Addition, we can deduce \(x \leq y\) from the premise \(x < y\).

Key Concepts

Understanding Inequality

An inequality is a mathematical statement that describes a relationship between two values. In our case, we are considering the inequality where one value, say x, is less than another value, y. This is written as \( x < y \).
When we talk about weakening an inequality, we mean making it less strict to include a broader range of situations.
For example, if we know that \( x < y \), we often want to deduce that \( x \leq y \), which means x is either less than or equal to y.
This involves understanding that \( x \leq y \) is shorthand for the disjunction \( x < y \) or \( x = y \). Understanding that this weakened inequality still holds true helps in many mathematical arguments and proofs.

Role of Disjunctive Addition

Disjunctive Addition is a rule of inference used in logic and proofs. It states that if a statement is true, you can add any other statement to it using 'or' to form a new true statement.
For instance, if \( x < y \) is true, we can add the statement \( x = y \) (even if it might not be true) to it using 'or', forming \( x < y \vee x = y \). This disjunction remains true because \( x < y \) was already true.
Therefore, Disjunctive Addition helps in changing strict inequalities (like \( x < y \)) into less strict ones (like \( x \leq y \)), which is often useful in proofs where stronger restrictions are not necessary.

Understanding Rules of Inference

Rules of inference are the foundational principles that guide logical reasoning in mathematics. They help us draw valid conclusions from given statements.
Some common rules of inference include Modus Ponens, Modus Tollens, and Disjunctive Addition. In our case, we use Disjunctive Addition to infer \( x \leq y \) from \( x < y \).
By understanding these rules, students can construct valid arguments and proofs. In mathematics, using correct rules of inference ensures that each step logically follows from the previous one, maintaining the integrity of the argument.