# Biconditionals

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##### Intros

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##### Examples

###### Lessons

**Forming the Biconditional**

Write a biconditional using the given conditionals:- If today is Monday, then yesterday was Sunday.

If yesterday was Sunday, then today is Monday - If he drives faster than 100 km, then he will get a ticket.

If he gets a ticket, then he has driven faster than 100 km. - If today is the weekend, then I do not go to school.

If I do not go to school, then it is the weekend.

- If today is Monday, then yesterday was Sunday.
- You are given 3 statements in symbolic form:

$p$: There is cream in milk

$q$: There is milk in coffee

$r$: There is cream in coffee **Truth value of Biconditionals**

Determine the truth value of the following biconditionals:- Let both $p$, $q$ and $r$ be statements.
- If $p$ is true, $q$ is true and $r$ is false, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?
- If $p$ is false, $q$ is true and $r$ is false, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?
- If $p$ is false, $q$ is false and $r$ is true, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?
- If $p$ is false, $q$ is false and $r$ is false, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?

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###### Topic Notes

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**biconditional**is a conjunction of a conditional and its converse. In symbolic form it would be:$(p \to q)$ ˄ $(q \to p)$

We can also write it as $p \leftrightarrow q$. In words, we connect $p$ and $q$ with "if and only if". A biconditional can be only true if both the conditional and converse is true. Here is a truth table of a biconditional.

$p$ | $q$ | $p \to q$ | $q \to p$ | $(p \to q )$ ˄ $(q \to p)$ | $p \leftrightarrow q$ |

T | T | T | T | T | T |

T | F | F | T | F | F |

F | T | T | F | F | F |

F | F | T | T | T | T |

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