As a Rule
As a Set of Ordered Pairs
Concept of function
We may imagine a function to be some sort of a box or
machine. When something is fed into the machine, there
will be an output. The output must be unique. This means
that no matter how many
A proposition may consist of other propositions. The truth
value of this proposition may change depending on the truth
values of the other propositions it is made of. The truth
values for all possible cases are displayed in tabular for
Reflexive, Symmetric, and Transitive
Definition. A binary relation on a set is called an
equivalence relation if it is reflexive, symmetric, and
Example. Let and
The table below gives the properties of the five relations.
Predicates and Quantifiers
Let be a non-negative integer. An -place predicate is a
statement containing variables. By fixing the values of the
variables in a predicate, it becomes a proposition.
Definition. A binary relation on a set is any subset of .
Example. Let . Then
, (2,1), (2,2), (2,3), (3,1), (3,2), (3,3)
By definition, a binary relation on is any subset of .
Therefore, the following are binary relations on the set :
Principle of Inclusion
Principle of Inclusion and Exclusion
Let , , , be arbitrary sets. Then
Exercise. Write the formula for using the principle of
inclusion and exclusion.
The red circle means that the elements in
Let us first discuss a new and more convenient notation for
membership in a binary relation. If is a binary relation on a
set , then the statement will be written as .
Example. Consider the binary relati
Definition. A binary relation on a set is reflexive if for
all , .
Example. Let . Consider the following binary relations
Among the three given relation, only and are reflexive.
Note that for any se