If P and Q are sentences, then the sentence P and Q is called the
conjunction
of P and Q, denoted by 
. For any statement
there are just two possible truth values, true (T) or false (F). If P and
Q are both true, then is true. If one or both of P and Q are
false, then is false. The truth table below defines the truth values
of for all possible truth value combinations of P and Q.
If P and Q are sentences, then the sentence P or Q is called the
disjunction
of P and Q, denoted by 
. In mathematics we use
an inclusive or. That is, is true when P is true, or Q is
true, or both are true. is false only when P and Q are false. the
truth table for is thus defined below.
| P | Q | |
| T | T | T |
| T | F | F |
| F | T | F |
| F | F | F |
| P | Q | |
| T | T | T |
| T | F | T |
| F | T | T |
| F | F | F |
| P |  |
| T | F |
| F | T |
A negation , or denial, of a sentence is formed in many ways. For example the negation of
P: 2 is rational.is represented by each of the following:
You need to realize that there are other symbols, besides 
, for
negations that are in common usage.
means 
means 
means 
If P and Q are sentences, the sentence
If P, then Qis denoted by
or
. We construct a truth table for just as for the
the other connectives 
, 
, and . However, the definition is
not at all obvious. Consider the sentence:
If I get an A in mathematics, then I will take the next course.Suppose a fellow student says this. When is the sentence true and when is it false? Let P denote the statement
I get an A in mathematicsand let Q denote the statement
I will take the next course.Consider the following four cases.
| P | Q | |
| T | T | T |
| T | F | F |
| F | T | T |
| F | F | T |
The sentence is called a conditional
with P the
antecedent and Q the consequent. In mathematics the conditional is
encountered in many forms. The following have the same meaning: