# What are m-placed relation symbols and function symbols?

In This Model Theory book, Weiss refers to "m-placed function symbols" and "m-placed relation symbols". Are these just supposed to be functions of $m$ objects and relations between $m$ objects? I'm having some trouble digesting this. I think I am having more trouble understanding the m-placed relations, though.

They're symbols that are intended to denote such functions and relations; they're not functions or relations themselves.

• Right, thanks. I'm just having a bit of trouble understanding multi-valued relations, though... it's essentially a way to show that $m$ objects are related under some operation? – adamcatto Jul 25 '16 at 17:35
• If $\underline{\text{R}}$ is, say, a 3-place relation symbol, then you can write formulas of the form $\underline{\text{R}}(x,y,z)$ (where $x,$ $y,$ and $z$ are variables, constant symbols, or other terms), and you can use those as subformulas of sentences or other formulas. None of these formulas mean anything specific until you have a model with (among other things) an actual 3-place relation $R$ that you will be using as the interpretation of the symbol $\underline{\text{R}}.$ In fact, the meaning of these formulas will be different in different models. – Mitchell Spector Jul 25 '16 at 17:45
• Oh I see. For some reason, I was under the impression that $R(x_1 \cdots x_n)$ was something like an equivalence class, under some binary relation $\sim$, such that for all $x_i$ and $x_j$, $x_i \sim x_j$. I understand now, though, thank you! – adamcatto Jul 25 '16 at 17:52
• I didn't have room to include this in my previous comment, but a 3-place relation $R$ in a model with universe $A$ is just a subset of $A \times A \times A,$ where we say that $R(x,y,z)$ is true iff $\langle x,y,z\rangle$ belongs to that subset. – Mitchell Spector Jul 25 '16 at 17:54
• Oh, that actually makes a lot of sense. Thanks a lot! – adamcatto Jul 25 '16 at 17:58

Just think of a relation as a collection of ordered $m$-tuples. Items $a1$, $a2$, ..., $am$ are related under the $m$-place relation R iff the tuple ($a1$, $a2$, ..., $am$) is in R. [they don't have to be related 'under some operation' to be related]

For example, there is the 3-place relation which I will call $Pyth()$ : the numeric triplet ($a$,$b$,$c$) is in $Pyth()$ iff $a+b=c$. When considering $Pyth()$ as a relation, we don't need to involve the '+' or '=' if we don't want to, we can just think about the triplets '(3,4,5)', '(5,12,13)', etc., by themselves (as a collection). Model Theory takes that perspective.

• Great explanation, thanks. – adamcatto Jul 25 '16 at 18:01

In Set Theory an $n$-ary relation on a set $S$ is some (any) subset of $S^n.$ And an $n$-ary function on $S$ is a function from $S^n$ into $S.$ In Model Theory an $n$-ary relation and $n$-ary function symbols are arbitrary objects that are "interpreted" by a model $\mathbb M=(M,...)$ as $n$-ary relations and $n$-ary functions on $M.$ The "..." in $\mathbb M=(M,...)$ is a "list" of the interpretations by $\mathbb M$ of all the relation symbols and function symbols of the language

• I took courses in set theory , and model theory under William Weiss at U. of Toronto. He had a difficulty which he said was embarrassing for a mathematician : An allergy to chalk. He had a large wheeled greaseboard. – DanielWainfleet Jul 25 '16 at 18:00
• ^ oh my, haha... – adamcatto Jul 25 '16 at 18:04
• What about using whiteboards instead? – user21820 Jul 26 '16 at 4:31
• @user21820 I think whiteboard is indeed the more accurate term for what he used – DanielWainfleet Jul 26 '16 at 4:51