# Biconditionals

### Biconditionals

#### Lessons

A 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
• Introduction
Biconditionals Overview:
a)
What are Biconditionals

• 1.
Forming the Biconditional
Write a biconditional using the given conditionals:
a)
If today is Monday, then yesterday was Sunday.
If yesterday was Sunday, then today is Monday

b)
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.

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

• 2.
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

a)
Write the conditional $(p$ ˄ $q)$ $\to$ $r$ in words.

b)
Write the converse$(p$ ˄ $q)$ $\to$ $r$ in words.

c)
Write the biconditional.

• 3.
Truth value of Biconditionals
Determine the truth value of the following biconditionals:
a)
$x = 5$. if and only if $x + 7 = 12$.

b)
If $x > 5$, if and only if $x > 2$.

c)
The triangle is an isosceles, if and only if two sides are equal.

d)
The angle is acute, if and only if the angle is less than 90° degrees.

• 4.
Let both $p$, $q$ and $r$ be statements.
a)
If $p$ is true, $q$ is true and $r$ is false, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?

b)
If $p$ is false, $q$ is true and $r$ is false, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?

c)
If $p$ is false, $q$ is false and $r$ is true, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?

d)
If $p$ is false, $q$ is false and $r$ is false, then what is the truth value of the biconditional $(p$ ˄ $q)$ $\leftrightarrow$$r$ ?