# a=b, b=c, therefore a=c -- what theorem is this? **URL:** \ **Category:** Factual Questions\ **Created:** [March 28, 2003, 8:35am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745 "2003-03-28T08:35:42Z")\ **Posts on this page:** 19\ **Page:** 1
**Author:** ![superfreakicus](https://avatars.discourse-cdn.com/v4/letter/s/ee7513/32.png) [@superfreakicus](https://boards.straightdope.com/u/superfreakicus)\ **Post date:** [March 28, 2003, 8:35am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/1 "2003-03-28T08:35:42Z")
thanks ---
**Author:** ![psychonaut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/psychonaut/32/4655_2.png) [@psychonaut](https://boards.straightdope.com/u/psychonaut)\ **Post date:** [March 28, 2003, 8:38am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/2 "2003-03-28T08:38:48Z")
Transitivity. ---
**Author:** ![psychonaut](https://sea3.discourse-cdn.com/straightdope/user_avatar/boards.straightdope.com/psychonaut/32/4655_2.png) [@psychonaut](https://boards.straightdope.com/u/psychonaut)\ **Post date:** [March 28, 2003, 8:39am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/3 "2003-03-28T08:39:58Z")
Oh, and it’s not a theorem; it’s more of a postulate. You can’t exactly prove it. ---
**Author:** ![Crowbar\_of\_Irony\_3](https://avatars.discourse-cdn.com/v4/letter/c/f08c70/32.png) [@Crowbar\_of\_Irony\_3](https://boards.straightdope.com/u/Crowbar_of_Irony_3)\ **Post date:** [March 28, 2003, 8:57am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/4 "2003-03-28T08:57:28Z")
How about the C theroem? int main { int a; int b; int c; a=b=c=0; } ---
**Author:** ![John\_Kentzel-Griffin](https://avatars.discourse-cdn.com/v4/letter/j/ccd318/32.png) [@John\_Kentzel-Griffin](https://boards.straightdope.com/u/John_Kentzel-Griffin)\ **Post date:** [March 28, 2003, 9:00am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/5 "2003-03-28T09:00:55Z")
Yeah, it’s not a theorem. In math, you start with axioms or postulates. These are assumed to be true without proof. Transitivity is assumed to hold for equality. In general an equivalence relation is one that has these properties:[list=1][_]Reflexive – a=a[_]Symmetric – If a=b, then b=aTransitive – If a=b and b=c, then a=c[/list=1] ---
**Author:** ![Achernar](https://avatars.discourse-cdn.com/v4/letter/a/e274bd/32.png) [@Achernar](https://boards.straightdope.com/u/Achernar)\ **Post date:** [March 28, 2003, 9:09am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/6 "2003-03-28T09:09:49Z")
So for instance, for people, “lives in the same country as” would be an equivalence relation. “Is at least as tall as” would not be an equivalence relation because it’s not symmetric. “Has been in the same room as” would not be an equivalence relation because it is not transitive. ---
**Author:** ![David\_Simmons](https://avatars.discourse-cdn.com/v4/letter/d/9de053/32.png) [@David\_Simmons](https://boards.straightdope.com/u/David_Simmons)\ **Post date:** [March 28, 2003, 2:37pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/7 "2003-03-28T14:37:53Z")
I think that this is one of Euclid’s “Common notions” or postulates. “Things that are equal to the same thing are equal to each other.” ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 28, 2003, 2:47pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/8 "2003-03-28T14:47:23Z")
Actually, the transitive property of equality _is_ a theorem. If you have reflexivity (for all a, a = a) and the principle of substitution (if t = s and P(t) is a statement about t, then P(s) has the same truth value as P(t)), you can derive symmetry (for all a and b, if a = b then b = a) and transitivity (as in the OP). But this is an advanced matter. 99% of the mathematicians out there regard it as a postulate. ---
**Author:** ![Urban\_Ranger](https://avatars.discourse-cdn.com/v4/letter/u/e9c0ed/32.png) [@Urban\_Ranger](https://boards.straightdope.com/u/Urban_Ranger)\ **Post date:** [March 28, 2003, 3:50pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/9 "2003-03-28T15:50:03Z")
The Principle of Substitution doesn’t look like math. It looks like logic. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 28, 2003, 3:52pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/10 "2003-03-28T15:52:31Z")
> [@](#): > > \*Originally posted by Urban Ranger \* > \*\*The Principle of Substitution doesn’t look like math. It looks like logic. \*\* You think there’s a difference? ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 28, 2003, 4:02pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/11 "2003-03-28T16:02:18Z")
Worse, ultrafilter’s definition of the principle of substitution uses the equality of t=s, which as DrMatrix mentioned is usually defined by using the idea of transitivity. So, it’d be tough to use that to derive transitivity. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 28, 2003, 4:14pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/12 "2003-03-28T16:14:15Z")
> [@](#): > > \*Originally posted by RM Mentock \* > \*\*Worse, ultrafilter’s definition of the principle of substitution uses the equality of t=s, which as DrMatrix mentioned is usually defined by using the idea of transitivity. So, it’d be tough to use that to derive transitivity. \*\* Actually, ‘=’ is an undefined symbol in that definition, which just happens to satisfy that property. I’ll give you a more complete answer later. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 28, 2003, 4:47pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/13 "2003-03-28T16:47:56Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*I’ll give you a more complete answer later. \*\* Not necessary. It’s fairly easy to take a group of axioms and their derived theorems, and change the designation–make a theorem an axiom and derive a former axiom from it. But that doesn’t make _everything_ a theorem, and nothing an axiom. Clearly, you have to start somewhere sometime. Context is key. ---
**Author:** ![raygirvan](https://avatars.discourse-cdn.com/v4/letter/r/ac91a4/32.png) [@raygirvan](https://boards.straightdope.com/u/raygirvan)\ **Post date:** [March 28, 2003, 5:01pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/14 "2003-03-28T17:01:02Z")
**David Simmons** : yes. It’s an algebraic equivalent of the first General Axiom from [Euclid’s ‘The Elements’](http://www.philosophy.leeds.ac.uk/GMR/hmp/modules/sl9899/euclid/elements.html). To Euclid, it was one of a number of self-evident notions on which further his further reasoning was based. ---
**Author:** ![Hobie\_the\_One](https://avatars.discourse-cdn.com/v4/letter/h/e99b99/32.png) [@Hobie\_the\_One](https://boards.straightdope.com/u/Hobie_the_One)\ **Post date:** [March 28, 2003, 5:02pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/15 "2003-03-28T17:02:47Z")
Actually, isn’t the transitive property just an application of substitution? since a=b we can replace any occurance of b with a and still have a true statement. ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 28, 2003, 5:11pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/16 "2003-03-28T17:11:56Z")
> [@](#): > > \*Originally posted by Hobie the One \* > \*\*Actually, isn’t the transitive property just an application of substitution? > > since a=b we can replace any occurance of b with a and still have a true statement. \*\* I think that’s what ultrafilter is saying, leaving equality undefined. This could get messy. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 28, 2003, 7:13pm UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/17 "2003-03-28T19:13:06Z")
Messy? You don’t know the half of it. What I should’ve written is this: For any binary relation R, if R(a, a) for all a, and R(t, s) implies that P(t) has the same truth value as P(s) for any statement P, then R(a, b) implies R(b, a) for all a, b, and R(a, b) and R(b, c) implies R(a, c) for all a, b, c. In such case, R has all the same properties as ‘=’, so that’s how we usually denote it. We do use ‘=’ to represent weaker relations–for instance, most equivalence relations do not have the principle of substitution. That’s why this is not just a rearrangement of axioms into theorems and vice versa. You can have reflexivity, transitivity, and symmetry without being able to substitute. Note that for any particular area, the concept of ‘a = b’ is not undefined, but the symbols are undefined until we’re talking about a particular area. Is this a little clearer? ---
**Author:** ![RM\_Mentock](https://avatars.discourse-cdn.com/v4/letter/r/e274bd/32.png) [@RM\_Mentock](https://boards.straightdope.com/u/RM_Mentock)\ **Post date:** [March 29, 2003, 12:08am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/18 "2003-03-29T00:08:50Z")
> [@](#): > > \*Originally posted by ultrafilter \* > \*\*Is this a little clearer? \*\* Seems to be just DrMatrix’s definition, in a different form, as opposed to using substitution to define equivalence. ---
**Author:** ![ultrafilter](https://avatars.discourse-cdn.com/v4/letter/u/3d9bf3/32.png) [@ultrafilter](https://boards.straightdope.com/u/ultrafilter)\ **Post date:** [March 29, 2003, 1:05am UTC](https://boards.straightdope.com/t/a-b-b-c-therefore-a-c-what-theorem-is-this/164745/19 "2003-03-29T01:05:31Z")
> [@](#): > > \*Originally posted by RM Mentock \* > \*\*Seems to be just DrMatrix’s definition, in a different form, as opposed to using substitution to define equivalence. \*\* No, because DrMatrix’s definition is not sufficient to imply substitution.