# Logic

## Equivalent Statements


### Learning Objectives

After completing this section, you should be able to:
1. Determine whether two statements are logically equivalent using a truth table.
2. Compose the converse, inverse, and contrapositive of a conditional statement
Have you ever had a conversation with or sent a note to someone, only to have them misunderstand what you intended to convey? The way you choose to express your ideas can be as, or even more, important than what you are saying. If your goal is to convince someone that what you are saying is correct, you will not want to alienate them by choosing your words poorly.
Logical arguments can be stated in many different ways that still ultimately result in the same valid conclusion. Part of the art of constructing a persuasive argument is knowing how to arrange the facts and conclusion to elicit the desired response from the intended audience.
In this section, you will learn how to determine whether two statements are logically equivalent using truth tables, and then you will apply this knowledge to compose logically equivalent forms of the conditional statement. Developing this skill will provide the additional skills and knowledge needed to construct well-reasoned, persuasive arguments that can be customized to address specific audiences.

### Determine Logical Equivalence

Two statements,  and , are logically equivalent when  is a valid argument, or when the last column of the truth table consists of only true values. When a logical statement is always true, it is known as a tautology. To determine whether two statements  and  are logically equivalent, construct a truth table for  and determine whether it valid. If the last column is all true, the argument is a tautology, it is valid, and  is logically equivalent to ; otherwise,  is not logically equivalent to .

### Compose the Converse, Inverse, and Contrapositive of a Conditional Statement

The converse, inverse, and contrapositive are variations of the conditional statement, 
    
    
 The table below shows how these variations are presented symbolically.

### Check Your Understanding


### Key Terms

1. logically equivalent
2. tautology
3. inverse
4. converse
5. contrapositive

### Key Concepts

1. Two statements  and  are logically equivalent if the biconditional statement,  is a valid argument. That is, the last column of the truth table consists of only true values. In other words,  is a tautology. Symbolically,  is logically equivalent to  is written as: 
2. A logical statement is a tautology if it is always true.
3. To be valid a local argument must be a tautology. It must always be true.
4. Know the variations of the conditional statement, be able to determine their truth values and compose statements with them.
1. The converse of a conditional statement, if  then , is the statement formed by interchanging the hypothesis and conclusion. It is the statement if  then .
2. The inverse of a conditional statement if formed by negating the hypothesis and the conclusion of the conditional statement.
3. The contrapositive negates and interchanges the hypothesis and the conclusion.
    
    
    
4. The conditional statement is logically equivalent to the contrapositive.
5. The converse is logically equivalent to the inverse.
6. Know how to construct and use truth tables to determine whether statements are logically equivalent.