---
#logic
## Definition
> [!tldr] Definition
> A **conditional statement** is a [[Propositions|proposition]] that has the form "If $A$, then $B
quot; where $A$ and $B$ are [[Propositions|propositions]]. The proposition $A$ in the "if" part is called the **hypothesis** of the conditional statement, and the proposition $B$ in the "then" part is called the **conclusion**.
>
> The conditional statement "If $A$ then $Bquot; is written in mathematical notation as $A \rightarrow B$.
Notes:
- Conditional statements need not literally be phrased as "if-then" statements; any phrasing that indicates a conclusion follows from a hypothesis is equivalent. For example, here are five different ways to phrase the conditional statement "If it's snowing, then it's cold":
- *Snowing implies that it's cold*.
- *It's cold whenever it's snowing.*
- *Snowing is a sufficient condition for it to be cold.*
- *Being cold is a consequence of it snowing.*
- *Being cold follows from it snowing.*
[See this article for more](https://math.stackexchange.com/questions/42042/different-ways-to-express-if-then).
* We consider the arrow in $P \rightarrow Q$ to be a [[connective]] that joins the [[Propositions|propositions]] $P$ and $Q$.
* The [[Truth tables|truth table]] for the conditional statement $P \rightarrow Q$ is:
| $P$ | $Q$ | $P \rightarrow Q$ |
| -- | -- | ---- |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
> [!important] When are conditional statements false?
> Conditional statements are false only in one case: When the hypothesis is true but the conclusion is false. (That is, the consequence does not follow from the condition.)
* The [[Negation|negation]] of a conditional statement is *not* another conditional statement, rather it is the proposition $P \wedge (\neg Q)$. This is because the truth table for $P \wedge (\neg Q)$ is
| $P$ | $Q$ | $P \wedge (\neg Q)$ |
| -- | -- | ---- |
| T | T | F |
| T | F | T |
| F | T | F |
| F | F | F |
And this has the exact opposite truth values as $P \rightarrow Q$.
## Examples
Here are examples of conditional statements with the hypothesis and conclusions indicated:
| Statement | Hypothesis | Conclusion |
| -------------------------------------------------------------------------- | -------------------- | ------------- |
| If Bob has pizza, he'll get sick. | Bob has pizza | Bob gets sick |
| If $x > 3$ then $x^2 > 9$. | $x > 3$ | $x^2 > 9$ |
| A running time longer than 3 minutes follows from the input being too big. | The input is too big | The running time is longer than 3 minutes |
Note the last example shows that the hypothesis of the conditional statement doesn't have to appear at the first of the sentence; it is merely the condition that triggers the response, and this can be phrased in different ways.
## Resources
<iframe src="https://player.vimeo.com/video/588372005?h=411ac35f76" width="640" height="360" frameborder="0" allow="autoplay; fullscreen; picture-in-picture" allowfullscreen></iframe>
<p><a href="https://vimeo.com/588372005">Screencast 2.3: Conditional statements</a> from <a href="https://vimeo.com/user132700952">Robert Talbert</a> on <a href="https://vimeo.com">Vimeo</a>.</p>
Other resources:
- [Textbook section on conditional statements](https://math.libretexts.org/Courses/SUNY_Schenectady_County_Community_College/Discrete_Structures/01%3A_Introduction_to_Writing_Proofs_in_Mathematics/1.01%3A_Statements_and_Conditional_Statements#:~:text=more%20formal%20definition.-,Definition,true%20whenever%20P%20is%20true.), with practice exercises