Comparing Transitivity and SubstitutionDate: 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/ |
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