Operators
We are particularly interested in combining propositions by operators.
Definition :
A compound proposition is a statement obtained by combining propositions with logical operators.
Conjunction
Disjunction
The disjunction operator is the binary operator which, when applied to two propositions
and
, yields the proposition “
or
”. The disjunction operator returns T when at last one of the two propositions
or
is true. The operator
is commutative[3].
|
|
|
|
T | T | T | T |
T | F | T | T |
F | T | T | T |
F | F | F | F |
Note :
More than two propositions can be joined using logical operators. In this instances, it is important to be careful about how they are grouped.
Example :
is true.
is false.
Negation
The negation operator is a unary operator. Sometimes in mathematics it's important to determine what the opposite of a given mathematical statement is. This is usually referred to as "negating" a statement. One thing to keep in mind is that if a statement is true, then its negation is false (and if a statement is false, then its negation is true).
|
|
T | F |
F | T |
Implication
In logic, implication is relationship between different propositions where the second proposition is a logical consequence of the first. The statement
is denoted by
. One way to think of the meaning of
is to consider it a contract that says if the first condition is satisfied, then the second will also be satisfied. We say,
implies
.
If
then
.
Truth Table of implication T
T
F
T
T
F
F
F
F
T
T
T
F
F
T
T
the converse of
is
. The contapositive of
is
.
Equivalence
Logical equivalence is the condition of equality that exists between two statements or sentences in propositional logic. The relationship between the two statements translates verbally into "if and only if." In mathematics, logical equivalence is typically symbolized by a double arrow
. The expression
means
.
|
|
|
|
|
T | T | T | T | T |
T | F | F | T | F |
F | T | T | F | F |
F | F | T | T | T |
Equivalent propositions
Two propositions are equivalent if they have identical truth tables.