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

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.

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.

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.

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.

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.

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.