Math Notes Identity Relations

 

In the following theorems, all relations are subsets of the cross-product BC, so an implicit assumption in all theorems is

 

      r AB

 

In some theorems there will be a reference to the universal set of discourse.  The universal set of discourse is denoted by U.

 

Any temporary variables introduced in the theorems are assumed to be distinct.  That is, the temporary variables do not as free variables in any other expression.

 

Axiom 1: a I b a = b

 

Theorem 1: dom.I.R = R

(prove LHS = RHS)

(reflexive property of =)

      dom.I.R = dom.I.R

(Definition of set equality)

      ("x|x dom.I.R x dom.I.R)

(Definition of dom.I.R)

      ("x|x dom.I.R x {a|($b|: a I b b R):a})

(definition of )

      ("x|x dom.I.R ($a|($b|: a I b b R):a=x))

(Axiom 1)

      ("x|x dom.I.R ($a|($b|: a=b b R):a=x))

(one point rule)

      ("x|x dom.I.R ($b|: x=b b R))

(trading)

      ("x|x dom.I.R ($b| x=b:b R))

(one point rule)

      ("x|x dom.I.R x R)

(Definition of set equality)

      dom.I.R = R

End of Proof

 

Theorem 2: I = I-1

(reflexive property of )

      ("a,b|:a =b a = b)

(Symmetry of =)

      ("a,b|:a =b b = a)

(Axiom 1)

      ("a,b|:a I b b I a)

( of I)

      ("a,b|:<a,b> I <b,a> I)

(definition of inverse)

      ("a,b|:<a,b> I <a,b> I-1)

(("x|z r:P) ("a,b|<a,b> r :P[z := <a,b>]))

      ("x|: x I x I-1)

(Definition of set equality)

     I = I-1

End of Proof

 

Theorem 3: ran.I.P = P

(Theorem 1)

      dom.I.P = P

(Theorem 2)

      dom.I-1.P = P

(Relations Part 2 Theorem 1)

      ran.I.P = P

End of Proof

 

 

References:

 

Dijkstra, Edsgar W. and Scholten, Carel S. 1990 Predicate Calculus and Program Semantics, New York, NY: Springer-Verlag

 

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.