# 23.3 Some useful operators

## 23.3.3 The material conditional

This one can be really confusing. The basic idea that we seek to express using the \saymaterial conditional is \sayif $A$, then $B$. Unfortunately natural language is often quite confusing, because when we say \sayif $A$, then $B$ many people, not at all unreasonably, interpret this to mean that if $A$ is false and $B$ is true then it is not at all true to say \sayif $A$, then $B$. However, in mathematics \sayif is a one-way street; all we are saying is that should $A$ be true, then $B$ is also true (for \sayif $A$, then $B$ to hold.

###### Definition 23.3.1

The \saymaterial conditional, i.e. the logical operator denoted by $\implies$ has the following truth table

 $A$ $B$ $A\implies B$ F F T F T T T F F T T T

###### Theorem 23.3.1

The (mathematical) natural language statement \sayif $A$, then $B$ corresponds to $A\implies B$ in logic.

###### Technique 23.3.1

To prove a statement in the form \sayif $A$, then $B$ or \say$A$ if $B$ one must

1. 1.

Assume that $A$ is true.

2. 2.

Show that therefore $B$ is also true.

Note that if $A$ is false it doesn’t matter what happens, as $A\implies B$ is true in this case.

## 23.3.4 The material bi-conditional

This is the operator which corresponds to the natural language statement \say$A$ if and only if $B$, which is to say that if $A$ is true, then $B$ must also be true (and vice versa). The basic idea is that $A$ and $B$ are concomitant; we cannot have one without the other.

###### Definition 23.3.2

We define the logical operator $A\iff B$ using the following truth table,

 $A$ $B$ $A\iff B$ F F T F T F T F F T T T