Math Notes – Identity Relations

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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:**

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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.