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Suppose that $$a^2+a+1=b$$ Suppose also that $a=5/4$. What makes it valid to substitute $5/4$ into the first equation? Is it because equality is transitive?

Git Gud
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user101939
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    It's by definition of equality and because in deductive systems there is a postulated rule that allows you to make such a substituion. – Git Gud Oct 19 '13 at 20:15
  • @GitGud, what do you mean by postulated rule? – user101939 Oct 19 '13 at 20:16
  • I mean it's something that is true axiomatically. It's like that because you define it like that. Why is it that $1=1$ is true? It's by definition of equality, we define it that way. – Git Gud Oct 19 '13 at 20:17

3 Answers3

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It's just a basic principle of first-order logic with equality that if $a = b$ and $P(b)$ for a formula $P$, then $P(a)$. It's probably the most important aspect of equality that usually "goes without saying" in mathematics, except in logic courses.

user43208
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The reason why you can make such a substitution is that $a$ and $\dfrac{5}{4}$ are precisely the same, just written differently. Here's a long-winded way of making the substitution you describe.

\begin{align*} a &= \frac{5}{4} & \\ a^2 &= \left(\frac{5}{4}\right)^2 &\text{squaring both sides}\\ a^2 + a &= \left(\frac{5}{4}\right)^2 + \frac{5}{4} &\text{adding the first two lines}\\ a^2 + a + 1 &= \left(\frac{5}{4}\right)^2 + \frac{5}{4}+1 &\text{adding $1$ to both sides}\\ b &= \left(\frac{5}{4}\right)^2 + \frac{5}{4}+1 &\text{using the fact that $a$ satisfies the given equation} \end{align*}

The right hand side is precisely what you obtain when you replace $a$ by $\dfrac{5}{4}$.

Michael Albanese
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There are at least two ways to understand equality.

One way to treat it is as a primitive logical identity sign: $a=b$ means that ‘$a$’ and ‘$b$’ are signs for the same thing. Since they represent one and the same object, any property that can be proved of the object represented by $a$ must also hold of the object represented by $b$, because they are the same object. In this case one automatically gets a general substitution principle: $a$ can be replaced by $b$ because they mean the same thing: they are names for the same object.

Another way to proceed is the way that ZF set theory does, and to take equality as a defined property. For example, ZF says that ‘$a=b$’ is an abbreviation for $$\forall x(x\in a\iff x\in b).$$

Two sets are defined to be equal if they contain the same elements. If one does this, substitution must be asserted axiomatically, or proved as a theorem. Without such a proof (trivial if substitution is an axiom) it might or might not hold. Transitivity is similar.

But the usual understanding in algebra is the first type: when we write something like ‘$x = \frac54$’ we mean that the name ‘$x$’ is a synonym for whatever object is denoted by $\frac54$.

MJD
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    This isn’t quite accurate. Whether a logical system takes equality as a primitive relation or a defined one, one almost always gives the substitutivity property as an axiom or rule of the system. In principle one certainly could have a system where substitutivity was a theorem, but that’s not what’s done in most presentations of ZF(C), or any other standard theory I know. – Peter LeFanu Lumsdaine Dec 19 '13 at 03:29