Conditional statements come in many different forms, as I am sure you've started to see in this Chapter. In mathematics, you've learned properties that allow you to write mathematical expressions in a different way. One example of this is the symmetric property that says "If $x=3$, then $3=x$". In logic, we have properties that are similar to those as well.
| Name | Symbolic | Words |
|---|---|---|
| Conditional | $$p \rightarrow q$$ | If p, then q. |
| Converse | $$q \rightarrow p$$ | If q, then p. |
| Inverse | $$\sim p \rightarrow \sim q$$ | If not p, then not q. |
| Contrapositive | $$\sim q \rightarrow \sim p$$ | If not q, then not p. |
Some of these expressions are logically equivalent to the others. This means they are logically the same argument and an alternative way to write an argument.
| p | q | Conditional $$p \rightarrow q$$ | Contrapositive $$\sim q \rightarrow \sim p$$ |
|---|---|---|---|
| T | T | T | T |
| T | F | F | F |
| F | T | T | T |
| F | F | T | T |
As we can see, the original conditional statement is logically equivalent to the contrapositive.
| p | q | Converse $$q \rightarrow p$$ | Inverse $$\sim p \rightarrow \sim q$$ |
|---|---|---|---|
| T | T | T | T |
| T | F | T | T |
| F | T | F | F |
| F | F | T | T |
In this one, we can see that the converse statement and the inverse statement are logically equivalent to each other.
Write a statement that is logically equivalent to "If I eat this giant cookie, then I will feel sick."
Since this is the conditional statement, a statement logically equivalent to the conditional is the contrapositive. This means we would need to flip and negate our p and q. This would give us:
$$\text{"If I do not feel sick, then I will not eat this giant cookie."}$$
Sometimes we need to be able to write a disjunction as a conditional, or a conditional as a disjunction. To do that, we would use the logically equivalent statements below:
$$p \vee q \equiv \sim p \rightarrow q$$
Find the conditional, converse, inverse, and contrapositive of the following statement: "Charity goes to the Mall or to TJ Maxx."
Our first step is to write it as a logically equivalent statement in terms of a conditional. From our rule above, we would negate the p. Then the p implies the q, giving us:
$$\text{If Charity doesn't go to the mall, then she will go to TJ Maxx.}$$
Now that we have our conditional statement, we can write the converse, inverse, and contrapositive.
| Form | Symbolic | Words |
|---|---|---|
| Converse | $$q \rightarrow p$$ | If Charity goes to TJ Maxx, then she will not go to the mall. |
| Inverse | $$\sim p \rightarrow \sim q$$ | If Charity goes to the mall, then she will not go to TJ Maxx. |
| Contrapositive | $$\sim q \rightarrow \sim p$$ | If Charity doesn't go to TJ Maxx, then she will go to the mall. |