## Question

Theorem1.5.1. If ,then .

## Solution

Proof. Assume . Then x, 2-x and 2+x are all nonnegative.

Therefore, the product of these terms is also nonnegative. Adding 1 to this product gives a positive number, so: Multiplying out on the left side proves that as claimed.                                                                                                             􏰐

## Learning way

• You should practice this, learn how to write a Proof . You may do some scratchwork while you’re trying to figure out the logical steps of a proof. Your scratchwork can be as disorganized as you like—full of dead-ends, strange diagrams, obscene words, whatever. But keep your scratchwork separate from your final proof, which should be clear and concise.
• This is your first CLASS on math, show your search skill while you meet Strange Concept，also I would write some necessary Concept for you.

## Homework

• Do you know about what nonnegative means?
• Please figure out why we know that the product of these terms is also nonnegative and how dare we use it without proving it.

## New Concept

### Proof Method

method

In order to prove that IMPLIES :

1. Write,“Assume .”

2. Show that logically follows.

### Logical Deductions

Logical deductions, or inference rules, are used to prove new propositions using previously proved ones.

A fundamental inference rule is modus ponens. This rule says that a proof of P together with a proof that P IMPLIES Q is a proof of Q.

Inference rules are sometimes written in a funny notation. For example, modus ponens is written:

Rule. When the statements above the line, called the antecedents, are proved, then we can consider the statement below the line, called the conclusion or consequent, to also be proved.

A key requirement of an inference rule is that it must be sound: an assignment of truth values to the letters P, Q, ..., that makes all the antecedents true must also make the consequent true. So if we start off with true axioms and apply sound inference rules, everything we prove will also be true.

There are many other natural, sound inference rules, for example:

Rule. Rule. On the other hand,

Non-Rule. is not sound: if P is assigned T and Q is assigned F, then the antecedent is true and the consequent is not.

As with axioms, we will not be too formal about the set of legal inference rules. Each step in a proof should be clear and “logical”; in particular, you should state what previously proved facts are used to derive each new conclusion.