Comparing Transitivity and Substitution

Date: 10/01/2003 at 10:03:44
From: Wade
Subject: Transitive Property and Substituition

What is the difference between transitive property and substitution?  
Can substitution be used in place of transitivity?  

Date: 10/01/2003 at 12:09:46
From: Doctor Peterson
Subject: Re: Transitive Property and Substituition

Hi, Wade.

The two are very closely related; the difference lies in their 
generality and their role in logic.  Often they are interchangeable, 
but not always.

Substitution is a "common-sense" concept: if two things are equal, 
then one can be put in place of the other and nothing will change. 
Essentially, it is part of the definition of "equal": two things are 
equal if and only if they can be substituted for one another.  It can 
be used to explain why, for example,

  a = b + 5  and  b = c  implies that  a = c + 5

The first statement here could be replaced by ANY statement about b; 
it is very general.

Transitivity is a little more formal; it is one of a set of 
properties (relexivity, symmetry, and transitivity) used to define 
the concept of "equivalence relation" (of which equality is one 
example).  It also has a more specific definition than substitution; 
it only applies when we have two equalities:

  a = b  and  b = c  implies that  a = c

This can be considered a special case of substitution, replacing b 
with c in the equation a = b.  So we could always use the term 
"substitution" if we wished; but we could not use the term 
"transitivity" in place of "substitution" in cases where the same 
quantity (b above) is not found alone on one side of each equation.

You can see why we call transitivity a "property of equality" (or, 
more generally, of an equivalence relation), but do not call 
substitution a "property" of anything in particular.  It is more 
general than that.

Here is one place where I commented on the relationship of these 
concepts:

  Isosceles Trapezoid Proof
  http://mathforum.org/library/drmath/view/55425.html 

See also

  MathWorld: Equivalence Relation
  http://mathworld.wolfram.com/EquivalenceRelation.html 

If you have any further questions, feel free to write back.

- Doctor Peterson, The Math Forum
  http://mathforum.org/dr.math/