23.3 Some useful operators
23.3.1 Logical and
23.3.2 Logical or
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 oneway 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.
Assume that $A$ is true.

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 biconditional
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 