# Notation for “Defined as proportional to”

If I say $$x$$ is defined as $$y+z$$ then I can say $$x := y+z$$. If I want to say $$x$$ is defined as proportional to $$y+z$$, then how can I say that? Would I say $$x :\propto y+z$$?

• I would say that strictly speaking there is no such symbol because, e.g., saying "$x$ is proportional to $y + z$" doesn't pin $x$ down---rather, it describes a property of $x$ shared by many other quantities. Depending on context, you could still frame this relationship as a definition by writing something like, "For a fixed parameter $a$ (to be determined later), set $x := a (y + z)$. – Travis Feb 18 at 23:30
• While it is perhaps a meaningless distinction, I would say "being defined as" is more a statement of identity while "being proportionate to" is more descriptive, that is it assigns a predicate to the variable (thinking in terms of first-order logic), as opposed to just a value. In my writing, I generally would add a statement about $x$'s identity before declaring any properties it might have. For example, I may say "Let $x$ be a number such that $x\propto y+x$". – Nico Feb 18 at 23:32
• Any two number are proportional. What the context here? – Somos Feb 19 at 0:01
• Context: "The MSE cost function is defined as $C(w, b) \propto \sum_{x}{||y(x)-a(x)||^2}$" – Shrey Joshi Feb 19 at 0:27
• I would not recommend using a special symbol for this. I think the quote in your comment is clear enough, given the context where "cost functions" like that arise. – Mark S. Feb 19 at 12:29

The symbol $$\propto$$ is indeed meaning that. It means that there exists $$k$$ a constant in a $$\mathbb{K}$$ field so that $$x=k\cdot (y+z)$$.
• I mean to ask, how do I say it is defined as that. I can always say that $x=y+z$ but it is better to say $x := y+z$. Or does no such notation exist? – Shrey Joshi Feb 18 at 23:25
• Prop to already fullfil this task. For example, you can write $PV \propto T$ – PackSciences Feb 18 at 23:25
• But your example does not say "$PV$ is defined to be proportional to $T$". In fact, $PV$ is defined to be the product of $P$ and $V$. I think you have not understood what @Shrey is asking for, which is a symbol for "defined to be the following function, up to a multiplicative constant" – Mark S. Feb 19 at 12:28