1. Lecture 43 Section 10.1 Wed, Apr 6, 2005
Relations on a Set
Lecture 43
Section 10.1
Wed, Apr 6, 2005

2. Relations on a Set
A relation from a set A to a set B is a subset of A  B.
A relation on a set A is a relation from A to A, i.e., a subset of A  A.

3. Examples: Relations on a Set
Define the relation < on the real numbers as x < y if x is less than y.
Define the relation  on the integers as a  b if a divides b.
Define the relation  on Z as m  n if m – n is even.
Define the relation  on the set of all statements as p  q if p  q  p.

4. Inverse Relations Let R be a relation from A to B.
The inverse relation, denoted R–1, is defined by (a, b)  R–1 if and only if (b, a)  R.

5. Examples: Inverse Relations
The inverse of the relation < on the real numbers is the relation >.
The inverse of the relation  on the integers is the relation “is a multiple of.”
The inverse of the relation  on Z is itself.
The inverse of the relation  on statements, defined by p  q  p, is the relation defined by p  q  p.

6. Inverse of p  q  p Theorem: If p  q  p, then p  q  q. Proof:
Suppose that p  q  p.
Then
(p  q)  p  p or p.
(p  p)  (q  p)  p  p.
T  (q  p)  T.
q  p  T.

7. Inverse of p  q  p (q  p)  q  T  q.
(q  q)  (p  q)  q.
F  (p  q)  q.
p  q  q.
p  q  q.