Math Notes – Identity Relations

 

A member of the Identity Relation has the property that the first member of the ordered pair is is the same as the second member of the ordered pair.

 

Axiom 1: a I b º a = b

 

While the Identity Relation is a relatively simple relation with more or less obvious properties, the Identity Relation occurs frequently enough in many different relation-related and set-related theorems that a treatment of the Identity Relation is valuable.

 

The first Identity theorem asserts that the domain of a result is the result.

 

Theorem 1: dom.I.R = R

 

The Identity Relation is its own inverse.

 

Theorem 2: I = I^-1

 

As the domain of a result is the result, the range of an initial state is the initial state.

 

Theorem 3: ran.I.P = P

 

 

References:

 

Brettler, Eli 1999, Course notes and comments from M1090 and M2090 math and logic courses,  http://www.math.yorku.ca/Who/Faculty/Brettler/menu.html

 

Fejer, Peter and Simovici, Dan A., 1991, Mathematical Foundations of Computer Science, New York, NY: Springer-Verlag

 

Ganong, Rick 1999, Course notes and comments from M1090 and M2090 math and logic courses, http://www.math.yorku.ca/Who/Faculty/Ganong

 

Gries, David and Schneider, Fred B., 1993, A Logical Approach to Discrete Math, New York, NY: Springer-Verlag

 

Hu Sze-Tsen, 1963, Elements of Modern Algebra, San Francisco: Holden-Day, Inc.