>[!summary] A conditional statement is a proposition which a statement is in the form "If A then B" (Although doesn't need to written in this form) > We call A the hypothesis and B the conclusion. > **Key equation:** > Conditional statement: $A \to B$ >[!info]+ Read Time ⏱ **3 mins** # Definition A conditional statement is a [[Propositions|proposition]] which is most often a statement in the form "**If** A **then** B" where A and B are propositions. Sometimes we call A the hypothesis and B the conclusion of a conditional statement.[^1] Mathematically, we write "If A then B" as $A \to B$ >[!note] Conditional statements don't always need to be in the form "If A then B". We know if a statement is a conditional statement if it implied that a conclusion follows from a hypothesis > **If** a wire carries current, then **it** produces a magnetic field (If A then B) can be written as: >- A write produces a magnetic field **whenever** it carries a current >- **All** current carrying wires produce a magnetic field >- The fact that a wire carries current **implies** that it produces a magnetic field >- Producing a magnetic field is the **consequence** of a wire carrying wire As a truth table, our conditional statements will look like this: | $A$ | $B$ | $A \to B$ | | ----- | ----- | --------- | | True | True | True | | True | False | False | | False | True | True | | False | False | True | >[!bug] When are conditional statements false? Conditional statements are only false in one case: when the hypothesis is true, but the conclusion is false The [[Negation|negation]] of a conditional statement is not another conditional statement; rather, it is the proposition $A \land \neg B$. Notice the difference between the negation of this conditional statement and without (above) as a truth table (are the opposite truth values of $A \to B$) | $A$ | $B$ | $A \land \neg B$ | | ----- | ----- | ---------------- | | True | True | False | | True | False | True | | False | True | False | | False | False | False | ## Examples Here are examples of conditional statements on a table. Notice the hypothesis and conclusion in each statement. | Statement | Hypothesis | Conclusion | | --------------------------------------------- | -------------------------- | --------------------- | | If a number is divisible by 2 then it is even | A Number is divisible by 2 | The number is even | | All humans are mortal | Someone is human | They are mortal | | Any student who studies passes their exams | Student who studies | They pass their exams | --- > 🧠 Enjoy this walkthrough? [Support Math & Matter](https://github.com/rajeevphysics/Obsidian-MathMatter) with a star and help others learn more easily. --- [^1]: Definition adapted from Dr. Robert Talbert